I think it all gets logged. Their dispatcher knows where they make each pick-up and drop-off and when -- for everyone's physical and legal protection. If a cab stopped there, there had to be a record, unless it only appeared to be a cab.
I can't speak on DC but in Chicago and NYC you get in and tell them where you want to go and they take you there. Other than turning on the meter they don't do anything else. At least this is the case when you hail a cab. If someone calls the cab company for a ride then the dispatcher records the info to ensure that someone gets there. I know you can flag a cab down in DC but I am not certain how they handle that aspect of it. A friend of mine drove a cab in Boston and he paid a "rental" fee for the cab for his shift and whatever he got above that was his profit. Of course he had to pay for gas. As far as I know, unless he was sent somewhere by a dispatcher there was no record (but I could be mistaken).