When I think of "install" I think of general purpose OSes which can install software from almost any source no questions asked, or use the native out of the box support for software installations.
The similar distinction exists with android and iOS, and is probably why the term is popular in those communities too.
If nothing else, the term sideload makes very clear on platforms with native appstores or locked down distribution channels (consoles, phones...) that the install did not come from the native channels. Installs from game discs or the xbox store are inherently different from developer mode software and using the same term "install" for both disguises this fact.