Feels like you’re proposing a distinction that isn’t universal, and literally there are words (on and off-site) that describe the concept better than the term self-hosted.

Additionally, just because some of the physical infrastructure (the hardware) isn’t in your control doesn’t make it any less self-hosted. I’m sure there are some that own their building and electrical infrastructure too. Meanwhile I’m a schmuck that has to rent and pay a utility provider for electrical. That’s not particularly far from datacenter access.