So, I feel like I narrowed it down to be something that has to do with the browser or the way I have the engine configured, rather than the device. Unless, for some reason both of my devices are emitting mouse events.
I wonder if mouse events are always emitted on the web, regardless of the device? It would be nice if someone could confirm.
This is actually intended behaviour because the Web, as a platform (not just for Godot) is a mess. Your browser will be translating your inputs into mouse events to make sure super old webpages work as well even on modern phones.