To me, the distinction comes down to: carpooling implies some kind of repeated/scheduled vehicle sharing (like coworkers coordinating a route to get to work at 9a every weekday, while ridesharing is much more ad-hoc/on-demand (like craigslist rideshares between Seattle and Portland to split the gas bill).
Definitely agree that Lyft/Uber aren’t really ride sharing in that sense though — the driver’s goal is to earn a fare on a trip he wasn’t going to take otherwise, while the original ride shares were a cost-reduction on a trip the driver was going to make anyway.
> carpooling implies some kind of repeated/scheduled vehicle sharing
In the Bay Area there's something called Casual Carpool; essentially it's just a series of locations you can go to around work-travel hours, and people who have cars and are willing to pick up people going to the same area will randomly show up and pick people up. No scheduling or prior arrangements aside from the locations. (And it's been around way longer than Uber/Lyft/etc.)
Definitely agree that Lyft/Uber aren’t really ride sharing in that sense though — the driver’s goal is to earn a fare on a trip he wasn’t going to take otherwise, while the original ride shares were a cost-reduction on a trip the driver was going to make anyway.