Did you download and install the jsMath-image-fonts.zip file? This is required for the image-mode fallback methods to work properly when the TeX fonts are not installed on the computer running the browser.
It may be that you have the TeX fonts available on your computer, but not on the other ones you tested. In that case, jsMath would not be able to load the image font definitions, and would fail to run. That would account for the behavior you are seeing.
If that's not it, would you be willing to set up a userid for me to try it out on your Moodle site? I can often figure out the problem more easily that way. You can send me the information via the Moodle message system rather than posting a URL publicly, if you wish.