Confused about modifier “no” when paired with “kitsune”/”gitsune”

On the topic of "no" as a modifier, I am specifically curious how that applies to things such as "kyuubi-no-kitsune", and what "kyuubi" technically means, on its own.

Additionally, I am curious what "no-kitsune" would mean in other contexts, and where it might be appropriate to modify it to "no-gitsune".