I know about in-the-sky.org and its online planetarium feature, but that's from the Earth's perspective. Is there a planetarium (preferably online but if not, that's fine) where the observer is on another celestial body, like the Moon or Mars?
Yes, the free software Stellarium (also Wikipedia) can do that. It has a list of celestial bodies and you select which one you want to be on. When you open Stellarium go to Location Menu (or the button F6 for Windows OS) and that is where you set your viewing point.
There might be additional options in Where can I find/visualize planets/stars/moons/etc positions?
JPL’s Solar System Simulator lets you select from a list of various targets as well as viewpoints.
Another simulator allowing to travel and observe from different locations is Celestia.
It is not online, installation needed.