Yes, and the supervisor also pays a fraction (often 50%+!) of any incoming grant money to “overhead” — to the institution, for lab space, staff, operations, etc

Yes, and the supervisor also pays a fraction (often 50%+!) of any incoming grant money to “overhead” — to the institution, for lab space, staff, operations, etc. Why some of this money is categorized as “tuition” and other money as “overhead” is at the root of the question I think.