We have to find a way to propagate INSIDE_EMACS to ssh sessions, otherwise this will not work.

Can you please explain the problem a bit more? Once we're connected with a remote server via ssh then this terminal session runs with the config of the remote server anways, no?

Yes, if I have these configs on my remote servers, they will not work if INSIDE_EMACS is not defined. So, if we want to modify the prompt (and add the other functions) only in vterm. We have to somehow make sure that INSIDE_EMACS is set to something also when we are ssh-ed somewhere.

Would this do the trick? https://superuser.com/a/702751

Unfortunately, that solution relies on server-side configurations that often cannot be controlled by the user. A workaround would be to pass the information inside an LC_* variable, but this is not very satisfying. Some servers block these too. The only one that always should go through is TERM, but changing it may lead to other problems.