The opposite of self-hosted would be managed service.

You run it yourself at your own location however you want it

Vs

Someone runs it for you at their location. However the want it

VPS is someone loans you a VM at their location that you run yourself however you want to.

It’s still relevant to self-hosted because you still have to do all the work, you were just using their network, power, air conditioning, hardware and fire suppression. You’re still in the hook for installs and patches, configuration, and software issues.