I avoided "local" because the name could technically be claimed by both types: "It's called -local- because it's local to a particular zone" vs "It's called -local- because it's local to my current (local) zone."
So I went with floating vs fixed because it feels less prone to ambiguity. "Fixed" is fixed to a particular zone, and "floating" floats along with me as my current zone changes.
And fixed time I would call local time. Why call it fixed? Because it's tied to a location?