I don’t think it’s a “warp drive”.
It’s a “warp stasis” bubble or field: you’ve effectively suspended (or considerably slowed) the people inside to preserve them for the future — stasis.
I think you’re running into objections because while it is warp technology, it’s not a drive in that it’s not providing atypical propulsion (ie, Star Trek warp drives, Star Wars hyperspace drives, Babylon 5 subspace jump drives, etc).
You’re correct that we’re trying to define “warp drive”, but each of “warp” and “drive” are used independently — and your usage of “warp drive” doesn’t respect their independent usage. That you have a 1:1000 slowdown in perceived time to real time as your example suggests at some level you recognize this isn’t a drive.
I think warp stasis is a super cool technology all on its own, though.