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I tend to draw a distinction - side loading usually infers a supported but not mainstream way of installing applications - this xbox for example cannot side load without you paying a small fee to enable the developer mode, and the vast majority of software will be obtained via retail discs or the Xbox store. It's not a generic "install" mechanism native to the out of the box experience for the console - you have to do some extra work for this avenue to open.

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.