You have defined installing to be specifically from play store and sideloading as everything except it.

Google isn't trying to prevent installing, just sideloading works in this sentence because of what you have already defined but you are using this sentence in defense of that....

As OP stated, installing can mean on debian as an example, installing from both apt or either tarballs. Both are valid installations

So it is the same for google/android as well yet google is trying to actively prevent one part of the installing or make it really extremely hard to do so.

It is a dangerous precedent. And I would say that it severely limits what you mean by installing.

I got an PC, and I got internet connection, usually it isn't trying to prevent what I install if I am on linux.

Yet I am on android and earlier it used to do the same but now its a slippery slope where it either requires me to use adb or keep another device at me at all times if I ever want to install software on it.

Not because its not that these phones can't do it, In fact that they already do but they are removing it, simply because they can.