All this terminology is muddied and often misused. The whole term is supposed to refer to getting access to the root user, which is definitely an OS-level thing. One thing people often don't remember also is that gaining root on your phone and unlocking the bootloader are not only different things, but entirely separate things. You can gain root in your vendor ROM without being able to unlock the bootloader (common case on Amazon tablets), and you can install LineageOS in your phone after unlocking the bootloader but you still won't have root access in the OS unless you do an extra step, setting up Magisk or similar. You also have to re-set-up Magisk after every LineageOS update, which often means once a week if you do all the OTA updates in a timely manner. Gaining root is usually the easier thing and the less useful thing. It doesn't help much with getting a nearer-to-AOSP experience. You can remove some apps but there are still limits to what can be done without just flashing a different ROM with less crap in it.

When picking a phone I like to see if it's on the LineageOS devices page (you bring up a good point that different variations of the same model aren't equal, definitely watch out for that), as that means both that the bootloader is unlockable and that someone else is already maintaining a ROM for it, and hopefully will for years to come. If I just go after the shiny new hardware, chances are the bootloader is locked and it will never be unlocked, generating e-waste.