There is no such thing as a stable orbit “at sufficient distance to not be influenced by” gravity.
There’s no such thing as a sufficient distance.
I guess certain multi-body situations like Lagrange points might make it debatable about which "direction" you're falling though.
The GGP didn't say zero-g "is" falling. They said it's "the same as" falling. Which, for purposes of this discussion, it is, for the reason I gave--the key common property is being weightless, i.e., free falling.
Note, btw, that "free-fall" does not necessarily imply "downward acceleration". It just means "weightless". You could be weightless, in free fall, far out in deep space well away from all gravitating bodies, so that there is no well-defined notion of "downward acceleration" in your vicinity.
Regarding your second point, I think everywhere in the universe has some direction of the gravity field, however weak it is, and that would be the downward direction. But yea in some ideal place with exactly zero gravity, there's that special case.