I work at a university data center and we host open weight models (as usual, this is my private opinion). I would consider open weight something you can take and self-host for free, also commercially and also that you can use for training and research. Anything else is just, I don't know, just "usable for free in certain cases"?

Open source or source available is something else entirely. To consider something source available I would require the actual training data to be available too. That is exceedingly rare.