Joscha sorry assumed you had Moodle access. But glad you got there in the end.
I think you're right on the documentation level. It isn't of course LTi as a standard (which is fairly flexible) but how each provider and consumer implement it. Hopefully as more people start using LTi it will improve.
Sadly of course this is the open source world. And as you point out many users are not well equipped to handle technical issues and even if they are like yourself they can often only see one end of the pipe.