For the first point, find any PC without a dedicated GPU, like your budget HP special, complete with 4-8GB of total RAM, and try to run a new PC game like Alan Wake 2 - which won't even launch the game without a dedicated GPU, let alone be playable. So not a PC then, it doesn't play Alan Wake 2. Or you could look at something like the MacBook Neo I guess, that certainly is incapable of playing a wide variety of games, hundreds of thousands of games I'd imagine, and still we'd probably call it a PC.
For the second point, running PC games, natively wasn't mentioned in the requirement. If you add that in, depending on how you add it, once again, there are many PCs (macs, etc) which would be incapable of running a PC game like Alan Wake 2 natively, so I guess they're not PC either. Or if you want to define it as a PC if it can run any PC game at all natively, that really begs the question of what counts and doesn't count.
Because I'm not sure many PCs could run older PC games natively, especially DOS ones, anymore. Without an emulator. Which is the same thing as my Android point, the phones can run lots of PC games with emulators afaik.