I don't mind the term "own rental", but I think "own rental" has kind of gone from one thing to a broader meaning of "own rental". Originally, I viewed an "own rental" as a guy that was injured/not available, but coming back later in the season/near the deadline. So instead to trading for a true rental at the deadline, you are adding this player that is coming back from injury that you already have on your roster, thus "own rental". I've heard RBA use "own rental" that way a few times in the past and probably what Negan is referring to with his objection to the term "own rental".
More broadly, "own rental" is now used pretty much for anyone you might lose to UFA in the off-season even though that guy isn't really a 'rental' in the sense that most people call guys picked up at the deadline a rental. For example, even though Pesce has been with Carolina for 8 straight seasons (thus not really a "rental"), some call him an "own rental" simply because he might leave as a UFA at the end of the season. Possibly losing him for nothing is the reason he's called an "own rental".
What's the point of this post about "own rentals"? Maybe it's just to see how many times I can use the term "own rental" in a post.