I happen to remember "sideload" as a term of art for some online file locker sites to mean saving it to your cloud drive instead of downloading it to your computer. A cool usage, but it never caught on.
I think nomenclature as it exists in the PC software universe is closest in spirit on all fronts, in describing running software as, well, running software, and describing installing as installing. While a little conspiratorial in tone they're not wrong that "sideload" pushes the impression that controlling what software you run on your phone should be understood as non-default.