Since you're being pedantic, I think we should call it what it is: Livery service. Taxis, at least in my municipality, are the only vehicles legally permitted

Since you're being pedantic, I think we should call it what it is: Livery service. Taxis, at least in my municipality, are the only vehicles legally permitted to pick passengers up off the street, aka only taxis can be hailed in person. Uber and Lyft cannot do this (where I live), and that is the differentiator.