No one drives a self-driving car. If I ride in one that I own, are you still going to call it a taxi? If I am at work and call the one that I own to drive me on a short door-to-door trip to grab lunch, will you call it a taxi then? What if I share ownership with a neighbor? How about 10,000 neighbors?
I understand that the establishment will always attempt to maintain control, but I think the situation is too different and complicated for the taxi establishment to overcome. I think that Google Ventures agrees.
Your argument makes no sense. You said that "Uber with self-driving cars" should be considered a car rental service, not a taxi service, so why bring up examples like "If I ride in one that I own." Uber is clearly a taxi service. It clearly does taxi service like things, and that's true whether the cars are driven by people or by machine (and from the perspective of the customer, the distinction is mostly irrelevant).
Whether a taxi service with self-driving cars should be subject to taxi regulations is a separate issue. Some of the concerns that led to taxi regulations are no longer relevant to a self-driving taxi (e.g. people picking up tourists and robbing them). Some of the concerns that led to taxi regulations are still relevant to self-driving taxis (e.g. companies focusing coverage on high-demand areas and ignoring poorer areas).
It makes no sense today, but it will make sense in 5 ~ 10 years. I bring up individual ownership of self driving cars because that is the environment in which the debate will happen, not the current one in which no self-driving cars exist. What does a "taxi" service look like when Hertz can deliver a self-driving car to my door? How is it any different from what we call a taxi today? Hence, either every self-driving car will be called a "taxi", or none of them will. Or, there needs to be a clearer differentiator between them.
Whether the current service that Uber provides is a taxi or not is irrelevant to the future debate involving cars with no drivers, unless those cars have a fare printed on the side and a light on the roof. It's likely that the taxi lobby will push for NYC to charge every car in the city a non-linear congestion fee unless you have a medallion.
