I look at it a bit more simply. If I feel like I benefited from or was inspired by someone else’s specific work, I like to give them credit. I think it promotes the positive nature of remix culture, and hope that others would do the same for me. It also helps others find both versions and choose which is best for them.
I don’t think I’ve ever found myself debating whether or not something is a remix. When I go to publish, I know whether or not I’ve built my work on top of someone else’s.
As for your scenario, this is telling:
It sounds like your starting point was the other model. From my perspective, that makes your model a remix.