Another version that makes use of an ASCIIMath-like syntax would also be helpful. The ASCIIMath philosophy would be to render each symbols by at most two ascii expressions: the LaTeX name (if it is short and memorable, without leading backslash) and a short name that resembles the shape of the symbol (if possible, else no second name). The desmos.com graphing calculator is based on MathQuill and uses (a restricted form of) such input.
\mathrm: $\mathrm{abc123@!}$
\mathit: $\mathit{abc123@!}$ does it work?? cf. $abc123@!$
\mathbf: $\mathbf{abc123@!}$
\mathsf: $\mathsf{abc123@!}$
\mathtt: $\mathtt{abc123@!}$
\mathcal, \mathfrak: not available in MathQuill??
\underline: $\underline{xyz}$, \underline{\sqrt x}: $\underline{\sqrt x}$
\overline{xyz}: $\overline{xyz}$, \overline{\sqrt x}$\overline{\sqrt x}$
SubSup x_1^2: $x_1^2$, x_{1}^{2}: $x_{1}^{2}$
subscript x_11 $x_11$, x_{11}: $x_{11}$, x\subscript11: $x\subscript11$
superscript x^12: $x^12$, x^{12}: $x^{12}$, x\superscript12: $x\superscript12$, x\supscript12: $x\supscript12$
\frac xy: $\frac xy$, \dfrac xy: $\dfrac xy$, \cfrac xy: $\cfrac xy$, \fraction xy: $\fraction xy$
xy\over uv: $xy over uv$, x{y\over u}v: $x{y over u}v$ only for live fractions
xy/uv: $xy/uv$, x{y/u}v: $x{y/u}v$ only for live fractions
\sqrt{x^2}: $\sqrt{x^2}$, \sqrt[n]{x^n}: $\sqrt[n]{x^n}$, root and bar have small gap in Firefox
(), [], \left\{\right\}, \langle, \rangle: $(), [], (], \left\{\right\}, leftlangle rightrangle$, \left{\right} need to be matched, \{\},\langle,\rangle don't work??
& lang; $⟨$, & rang; $⟩$,
pipe |: $|x$ works unmatched
x\text{ hi }y: $x\text{ hi }y$
x\textnormal{ hi }y: $x\textnormal{ hi }y$
x\textrm{ hi }y: $x\textrm{ hi }y$
x\textup{ hi }y: $x\textup{ hi }y$
x\textmd{ hi }y: $x\textmd{ hi }y$
a \emph{all}b: $a \emph{all}b$, a \em{all}b: $a \em{all}b$ why do \em??
a \italic{all}b: $a \italic{all}b$, a \italics{all}b: $a \italics{all}b$ ??
x\textit{ hi }y: $x\textit{ hi }y$
x\textbf{ hi }y: $x\textbf{ hi }y$, x\bold{ hi }y: $x\bold{ hi }y$, x\strong{ hi }y: $x\strong{ hi }y$,
x\textsf{ hi }y: $x\textsf{ hi }y$, x\sf{ hi }y: $x\sf{ hi }y$
x\texttt{ hi }y: $x\texttt{ hi }y$, x\tt{ hi }y: $x\tt{ hi }y$
x\textsc{ HIhi }y: $x\textsc{ HIhi }y$
x\uppercase{ HIhi }y: $x\uppercase{ HIhi }y$
x\lowercase{ HIhi }y: $x\lowercase{ HIhi }y$
\binom nk: $\binom nk$
n\choose k: $n choose k$ does it work?
\vector?
\f: $\f$ ?? why?
prime x': $x'$, x'': $x''$, x''': $x'''$
@: $@$, \&; $\&$, \%: $\%$, %: $%$
\alpha: $\alpha$
\beta: $\beta$
\gamma: $\gamma$
\delta: $\delta$
\zeta: $\zeta$
\eta: $\eta$
\theta: $\theta$
\iota: $\iota$
\kappa: $\kappa$
\mu: $\mu$
\nu: $\nu$
\xi: $\xi$
\rho: $\rho$
\sigma: $\sigma$
\tau: $\tau$
\chi: $\chi$
\psi: $\psi$
\phi: $\phi$ //W3C or Unicode? [PJipsen: Comments from mathquill.js]
\phiv: $\phiv$ //Elsevier and 9573-13
\varphi: $\varphi$ //AMS and LaTeX
\epsilon: $\epsilon$ //W3C or Unicode?
\epsiv: $\epsiv$ //Elsevier and 9573-13
\varepsilon: $\varepsilon$ //AMS and LaTeX
\piv: $\piv$ //W3C/Unicode and Elsevier and 9573-13
\varpi: $\varpi$ //AMS and LaTeX
\sigmaf: $\sigmaf$ //W3C/Unicode
\sigmav: $\sigmav$ //Elsevier
\varsigma: $\varsigma$ //LaTeX
\thetav: $\thetav$ //Elsevier and 9573-13
\vartheta: $\vartheta$ //AMS and LaTeX
\thetasym: $\thetasym$ //W3C/Unicode
\upsilon: $\upsilon$ //AMS and LaTeX and W3C/Unicode
\upsi: $\upsi$ //Elsevier and 9573-13
//these aren't even mentioned in the HTML character entity references
\gammad: $\gammad$ //Elsevier
\Gammad: $\Gammad$ //9573-13 -- was this a typo in the reference (see above)
\digamma: $\digamma$ //LaTeX
\kappav: $\kappav$ //Elsevier
\varkappa: $\varkappa$ //AMS and LaTeX
\rhov: $\rhov$ //Elsevier and 9573-13
\varrho: $\varrho$ //AMS and LaTeX
//Greek constants, look best in un-italicised Times New Roman
\pi: $\pi$
\lambda: $\lambda$
//uppercase greek letters
\Upsilon: $\Upsilon$ //LaTeX
\Upsi: $\Upsi$ //Elsevier and 9573-13
\upsih: $\upsih$ //W3C/Unicode "upsilon with hook"
\Upsih: $\Upsih$ //'cos it makes sense to me
//other symbols with the same LaTeX command and HTML character entity reference
\Gamma: $\Gamma$
\Delta: $\Delta$
\Theta: $\Theta$
\Lambda: $\Lambda$
\Xi: $\Xi$
\Pi: $\Pi$
\Sigma: $\Sigma$
\Phi: $\Phi$
\Psi: $\Psi$
\Omega: $\Omega$
Algebra
$x+y$
$x-y$, unary minus $-x$
$\pm x$, $\mp x$
x*y: $x*y$, x\cdot y: $x\cdot y$, x\sdot y: $x\sdot y$
x=y: $x=y$, x & lt; y: $x \lt y$, x& gt;y: $x \gt y$ can't use lt, gt on keyboard
\sim: $\sim$
\cong: $\cong$
\equiv: $\equiv$
\oplus: $\oplus$
\otimes: $\otimes$
\times: $\times$
\div: $\div$, \divide: $\divide$, \divides: $\divides$
\ne: $\ne$, \neq: $\neq$
\ast: $\ast$, \star: $\star$, \loast: $\loast$, \lowast: $\lowast$
\therefor: $\therefor$, \therefore: $\therefore$
\cuz: $\cuz$, \because: $\because$
\prop: $\prop$, \propto: $\propto$
\asymp: $\asymp$, \approx: $\approx$
\lt: $\lt$
\gt: $\gt$
\le: $\le$, \leq: $\leq$
\ge: $\ge$, \geq: $\geq$
\isin: $\isin$, \in: $\in$
\notin: $\notin$
\ni: $\ni$, \contains: $\contains$
\notni: $\notni$, \niton: $\niton$, \notcontains: $\notcontains$, \doesnotcontain: $\doesnotcontain$
\sub: $\sub$, \subset: $\subset$
\sup: $\sup$, \supset: $\supset$, \superset: $\superset$
\nsub: $\nsub$, \notsub: $\notsub$, \nsubset: $\nsubset$, \notsubset: $\notsubset$
\nsup: $\nsup$, \notsup: $\notsup$, \nsupset: $\nsupset$, \notsupset: $\notsupset$, \nsuperset: $\nsuperset$, \notsuperset: $\notsuperset$
\sube: $\sube$, \subeq: $\subeq$, \subsete: $\subsete$, \subseteq: $\subseteq$
\supe: $\supe$, \supeq: $\supeq$, \supsete: $\supsete$, \supseteq: $\supseteq$, \supersete: $\supersete$, \superseteq: $\superseteq$
\nsube: $\nsube$, \nsubeq: $\nsubeq$, \notsube: $\notsube$, \notsubeq: $\notsubeq$, \nsubsete: $\nsubsete$, \nsubseteq: $\nsubseteq$, \notsubsete: $\notsubsete$, \notsubseteq: $\notsubseteq$
\nsupe: $\nsupe$, \nsupeq: $\nsupeq$, \notsupe: $\notsupe$, \notsupeq: $\notsupeq$, \nsupsete: $\nsupsete$, \nsupseteq: $\nsupseteq$, \notsupsete: $\notsupsete$, \notsupseteq: $\notsupseteq$, \nsupersete: $\nsupersete$, \nsuperseteq: $\nsuperseteq$, \notsupersete: $\notsupersete$, \notsuperseteq: $\notsuperseteq$
//sum, product, coproduct, integral
\sum: $\sum$, \summation: $\summation$
\prod: $\prod$, \product: $\product$
\coprod: $\coprod$, \coproduct: $\coproduct$
\int: $\int$, \integral: $\integral$
//the canonical sets of numbers
\N: $\N$, \naturals: $\naturals$, \Naturals: $\Naturals$
\P: $\P$, \primes: $\primes$, \Primes: $\Primes$, \projective: $\projective$, \Projective: $\Projective$, \probability: $\probability$, \Probability: $\Probability$
\Z: $\Z$, \integers: $\integers$, \Integers: $\Integers$
\Q: $\Q$, \rationals: $\rationals$, \Rationals: $\Rationals$
\R: $\R$, \reals: $\reals$, \Reals: $\Reals$
\C: $\C$, \complex: $\complex$, \Complex: $\Complex$, \complexes: $\complexes$, \Complexes: $\Complexes$, \complexplane: $\complexplane$, \Complexplane: $\Complexplane$, \ComplexPlane: $\ComplexPlane$
\H: $\H$, \Hamiltonian: $\Hamiltonian$, \quaternions: $\quaternions$, \Quaternions: $\Quaternions$
//spacing
\quad: $|\quad|$, \emsp: $|\emsp|$, \qquad: $|\qquad|$
\, \,, \: \:, \; \;, \! \! //these cause mathquill-textbox errors??
//binary operators
\diamond: $\diamond$
\bigtriangleup: $\bigtriangleup$
\ominus: $\ominus$
\uplus: $\uplus$
\bigtriangledown: $\bigtriangledown$
\sqcap: $\sqcap$
\triangleleft: $\triangleleft$
\sqcup: $\sqcup$
\triangleright: $\triangleright$
\odot: $\odot$
\bigcirc: $\bigcirc$
\dagger: $\dagger$
\ddagger: $\ddagger$
\wr: $\wr$
\amalg: $\amalg$
//relationship symbols
\models: $\models$
\prec: $\prec$
\succ: $\succ$
\preceq: $\preceq$
\succeq: $\succeq$
\simeq: $\simeq$
\mid: $\mid$
\ll: $\ll$
\gg: $\gg$
\parallel: $\parallel$
\bowtie: $\bowtie$
\sqsubset: $\sqsubset$
\sqsupset: $\sqsupset$
\smile: $\smile$
\sqsubseteq: $\sqsubseteq$
\sqsupseteq: $\sqsupseteq$
\doteq: $\doteq$
\frown: $\frown$
\vdash: $\vdash$
\dashv: $\dashv$
//arrows
\longleftarrow: $\longleftarrow$
\longrightarrow: $\longrightarrow$
\Longleftarrow: $\Longleftarrow$
\Longrightarrow: $\Longrightarrow$
\longleftrightarrow: $\longleftrightarrow$
\updownarrow: $\updownarrow$
\Longleftrightarrow: $\Longleftrightarrow$
\Updownarrow: $\Updownarrow$
\mapsto: $\mapsto$
\nearrow: $\nearrow$
\hookleftarrow: $\hookleftarrow$
\hookrightarrow: $\hookrightarrow$
\searrow: $\searrow$
\leftharpoonup: $\leftharpoonup$
\rightharpoonup: $\rightharpoonup$
\swarrow: $\swarrow$
\leftharpoondown: $\leftharpoondown$
\rightharpoondown: $\rightharpoondown$
\nwarrow: $\nwarrow$
//Misc
\ldots: $\ldots$
\cdots: $\cdots$
\vdots: $\vdots$
\ddots: $\ddots$
\surd: $\surd$
\triangle: $\triangle$
\ell: $\ell$
\top: $\top$
\flat: $\flat$
\natural: $\natural$
\sharp: $\sharp$
\wp: $\wp$
\bot: $\bot$
\clubsuit: $\clubsuit$
\diamondsuit: $\diamondsuit$
\heartsuit: $\heartsuit$
\spadesuit: $\spadesuit$
//variable-sized
\oint: $\oint$
\bigcap: $\bigcap$
\bigcup: $\bigcup$
\bigsqcup: $\bigsqcup$
\bigvee: $\bigvee$
\bigwedge: $\bigwedge$
\bigodot: $\bigodot$
\bigotimes: $\bigotimes$
\bigoplus: $\bigoplus$
\biguplus: $\biguplus$
//delimiters
\lfloor: $\lfloor$
\rfloor: $\rfloor$
\lceil: $\lceil$
\rceil: $\rceil$
\slash: $\slash$
\opencurlybrace: $\opencurlybrace$
\closecurlybrace: $\closecurlybrace$
//various symbols
\caret: $\caret$
\underscore: $\underscore$
\backslash: $\backslash$
\vert: $\vert$
\perp: $\perp$, \perpendicular: $\perpendicular$
\nabla: $\nabla$, \del: $\del$
\hbar: $\hbar$
\AA: $\AA$, \Angstrom: $\Angstrom$, \angstrom: $\angstrom$
\ring: $\ring$, \circ: $\circ$, \circle: $\circle$
\bull: $\bull$, \bullet: $\bullet$
\setminus: $\setminus$, \smallsetminus: $\smallsetminus$
\not: $\not$, \neg: $\neg$
\dots: $\dots$, \ellip: $\ellip$, \hellip: $\hellip$, \ellipsis: $\ellipsis$, \hellipsis: $\hellipsis$
\converges: $\converges$, \darr: $\darr$, \dnarr: $\dnarr$, \dnarrow: $\dnarrow$, \downarrow: $\downarrow$
\dArr: $\dArr$, \dnArr: $\dnArr$, \dnArrow: $\dnArrow$, \Downarrow: $\Downarrow$
\diverges: $\diverges$, \uarr: $\uarr$, \uparrow: $\uparrow$
\uArr: $\uArr$, \Uparrow: $\Uparrow$
\to: $\to$, \rarr: $\rarr$, \rightarrow: $\rightarrow$
\implies: $\implies$, \rArr: $\rArr$, \Rightarrow: $\Rightarrow$
\gets: $\gets$, \larr: $\larr$, \leftarrow: $\leftarrow$
\impliedby: $\impliedby$, \lArr: $\lArr$, \Leftarrow: $\Leftarrow$
\harr: $\harr$, \lrarr: $\lrarr$, \leftrightarrow: $\leftrightarrow$
\iff: $\iff$, \hArr: $\hArr$, \lrArr: $\lrArr$, \Leftrightarrow: $\Leftrightarrow$
\Re: $\Re$, \Real: $\Real$, \real: $\real$
\Im: $\Im$, \imag: $\imag$, \image: $\image$, \imagin: $\imagin$, \imaginary: $\imaginary$, \Imaginary: $\Imaginary$
\part: $\part$, \partial: $\partial$
\inf: $\inf$, \infin: $\infin$, \infty: $\infty$, \infinity: $\infinity$
\alef: $\alef$, \alefsym: $\alefsym$, \aleph: $\aleph$, \alephsym: $\alephsym$
\forall: $\forall$ thankfully mathquill has only one version of forall [PJ]
\xist: $\xist$, \xists: $\xists$, \exist: $\exist$, \exists: $\exists$
\and: $\and$, \land: $\land$, \wedge: $\wedge$
\or: $\or$, \lor: $\lor$, \vee: $\vee$
\o: $\o$, \O: $\O$, \empty: $\empty$, \emptyset: $\emptyset$, \oslash: $\oslash$, \Oslash: $\Oslash$, \nothing: $\nothing$, \varnothing: $\varnothing$
\cup: $\cup$, \union: $\union$
\cap: $\cap$, \intersect: $\intersect$, \intersection: $\intersection$
\deg: $\deg$, \degree: $\degree$
\ang: $\ang$, \angle: $\angle$
\ln: $\ln$
\lg: $\lg$
\log: $\log$
\span: $\span$
\proj: $\proj$
\det: $\det$
\dim: $\dim$
\min: $\min$
\max: $\max$
\mod: $\mod$
\lcm: $\lcm$
\gcd: $\gcd$, \gcf: $\gcf$, \hcf: $\hcf$
\lim: $\lim$
\sin: $\sin$
\cos: $\cos$
\tan: $\tan$
\sec: $\sec$
\cosec: $\cosec$, \csc: $\csc$
\cotan: $\cotan$, \cot: $\cot$
\sinh: $\sinh$
\cosh: $\cosh$
\tanh: $\tanh$
\sech: $\sech$
\cosech: $\cosech$, \csch: $\csch$
\cotanh: $\cotanh$, \coth: $\coth$
\asin: $\asin$
\acos: $\acos$
\atan: $\atan$
\asec: $\asec$
\acosec: $\acosec$, \acsc: $\acsc$
\acotan: $\acotan$, \acot: $\acot$
\asinh: $\asinh$
\acosh: $\acosh$
\atanh: $\atanh$
\asech: $\asech$
\acosech: $\acosech$, \acsch: $\acsch$
\acotanh: $\acotanh$, \acoth: $\acoth$
\arcsin: $\arcsin$
\arccos: $\arccos$
\arctan: $\arctan$
\arcsec: $\arcsec$
\arccosec: $\arccosec$, \arccsc: $\arccsc$
\arccotan: $\arccotan$, \arccot: $\arccot$
\arcsinh: $\arcsinh$
\arccosh: $\arccosh$
\arctanh: $\arctanh$
\arcsech: $\arcsech$
\arccosech: $\arccosech$, \arccsch: $\arccsch$
\arccotanh: $\arccotanh$, \arccoth: $\arccoth$