The \mathcal rendering is the default one that is used in LaTeX's amsart document style (the format that is used as the basis for MathJax's commands). Note that the document you refer to in your link uses the calrsfs package, which changes the calligraphic font to the one from the rsfs font. So that is why \mathcal is different in that example.
MathJax does include another script font, however, that is probably what you are looking for. You can use \mathscr to obtain it (this is a macro from the mathrsfs package).
If you want that to be the default, you can use
\let\mathcal=\mathscr
to make the change.