Here is some possible 'help'....
to get the emulator software installed on networked school PCs, this advice might help (rather than students self installing as you outlined) ..
You are getting the 'latest' emulator. Sorry, the 'clunky' emulator is what you get at present. MIT is working on providing a 'more modern' looking emulator..it might be a long while before it is implemented for technical reasons. (ie, it is not a trivial change) There are other, more modern emulators available that can be used with App Inventor (GennyMotion is one). These more modern emulators are available from third part vendors and may not be available for free use except on individual PCs so probably not practical in your school environment. What you see on your Mac book is a different emulator package...so you get a slightly different presentation).
You also said "we are getting the attached error messages when we try to use the Emulator we've installed.
When we click the button to update, nothing appears to happen... but then the Emulator won't even open" ... this might be happening because of how you installed the basic emulator program ("when students installed it" ... App Inventor might not be finding the emulator in the expected PC directory. I don't know, just a guess. See the first link for installing the emulator using 'silent install', installing that way might fix the issue.
Here is the directory App Inventor expects to see the aiStarter software (emulator files).
If the stuff is not there, App Inventor cannot find it. Sorry, the software MUST be in that directory. Where is when the student's 'install' it?
Hopefully, something here will help you. If not, tell us what you tried please.
Regards,
Steve