In general, it seems quite obvious that the peer review process does not filter all articles with good content but that are badly written.

Are there any specific websites/services that are typically used for this ?

It is not based on a false premise. Not all universities give a stipend, and some ask for a non negligible tuition.

question: although there is no formal distinction as you say, is there nevertheless an informal term for these two types? such as "departmental PhD student" vs "institutional PhD student", or something like that. These don't have to be official terms recognized in the program.