IIRC= If I recall Correctly

In the years before the first Partition Russia was in a potential weak spot: war with Turks, Sweden potentially adding to the troubles making Russia go into a defensive alliance with Denmark, and Austria standing on the side lines worrying about Russian success in the Balkans/Black Sea. So yeah I would say Prussia ended up getting something that it had wanted for quite a while (land connecting to East Prussia) but Russia managed to appease their neighbours while expanding against the Turks.

Poland was not all peaceful though as Russia faced revolts and had to deal with that. Later on Poland seemed to gain strength and Russia (or at least some factions) saw Poland as having outlived its role as a protectorate. IMO Poland might have been vassal of Russia but it did not mean Russia had all control nor that Austria and especially Prussia had no influence at all. I do think Austria wanted it the least but better to take something than leave it all to Prussia.