All of these would seem to be "your data", but when they are carefully only including certain aspects (like prompts) in their statements it starts to sound they want to hide something.
Why wouldn’t they claim ownership of the AI output? They likely already claim ownership of the “transformation” (AI training) of the (pirated) input data.
For example, at school they can have an agreement with Gemini, but the student / academic could have bought an individual pro subscription to any other model provider.
About researchers, lots of them are probably using personal plans that aren't even reimbursed by their institutions. I could ask Cordova's research institution (I MAY) but I wouldn't be surprised at all if that was the case.