I'd argue it's clear from the context. The mushroom near Earth is "Spacedock Earth", a sombrero next to Vulcan is "Spacedock Vulcan", a dumbbell next to Andor is "Spacedock Andor", etc. The hundreds of dockyards around Earth do not perform the function of "Spacedock Earth", so they don't get called that in traffic control communication. Instead, they get called things like "Spacedock Four", and even that is probably shorthand for "Spacedock Four of the San Francisco Yards, Center Orbit Cluster".
I'd prefer "Spaceport" for such a central facility, but perhaps Spaceports in Trek are surface installations. And the traffic control entity might be called Space Central; there's decidedly a "Vulcan Space Central" controlling traffic in the Vulcan home system in "Amok Time". The "Space Central" in "Miri" might also be a traffic control facility, but all we know about it is that it will send teachers and advisers for the children on Miri's planet - so perhaps it's in fact short for Space Central High School?
As for "Starbase", I'd actually find it really odd if Starbase 1 sat on Earth orbit. That's not "a base at stars", not from the Earthling viewpoint! Surely the first Starbase would be founded in some distant star system, to complement Earthbase (or whatever) and to extend Starfleet's logistical reach via a second hub.
Perhaps a bit down the road, in the name of equality, a Starbase 77 or something like that might be established in the Sol system (that is, preexisting facilities there would be given the name Starbase 77 to calm down the Andorian critics who hate the fact that their home system has to play host to Starbase 66, making it look like Andoria is some sort of final frontier or wasteland).
Timo Saloniemi