Yes, obviously, you own the output. Training on those outputs is a violation of the service agreement, and that’s separate from who owns the outputs.
That sentence seems to be in a blurry line between outright “ownership” and licensing.
This type of explanation leans towards the reality being you don’t own the outputs from Claude.
This kind of an explanation is like trying to be half pregnant.
I don't see them as having a leg to stand on with this in court; they can only cut off your access for egregious violations.