Traditionally “public” transportation has usually referred to those modes of transport with set routes, set fares and usually the infrastructure and service was/is funded from public coffers and/or managed by government for the public though not in all cases. It’s almost always been mass transportation as well.
If I were to use healthcare as an analogy, it would be the difference between being covered only by NHIF and being covered by a private health insurance company like say, Resolution Health. Note how we all pay into NHIF whether we use it or not…? Public.
By this definition, taking a cab is opting out of public transport. That’s why the fares are higher and you get to ride in private. Uber is basically a cab. In the traditional definition it is not “public”. It’s a private means of transport. It’s not very different from chartering a plane.