so my rating on lichess is higher for rapid (close to 900) than to blitz (close to 700), and I kind of wonder whether this is due to differences in the pool of people playing rapid vs blitz, or whether I am just a not as good at thinking fast.

I have played A LOT of blitz games in June & July and only recently really picked up rapid games. Overall I think I am not even making too much use of the additional time quite often in rapid as I am still somewhat used to the time constraint in blitz (e.g. I never lost due to time-out on a rapid game), so I don't think its actually me playing much better in rapid than in blitz