This is just becoming a silly definition war. I'm not going to even bother to define "PC", I think it's a meaningless marketing term these days for things that vaguely resemble the linage of the original IBM PC. But, I do think it is possible for all the things you listed to be "general purpose computers". Anything that computes is a computer and is in any way programmable in order run a wide variety different algorithms is general purpose. Hence, there is a lot of general purpose computers out there, even if it may not seem like it.
Now going back to the SteamBox, it is in fact both a video game console and a general purpose computer. If it even vaguely resembles a vague concept of a PC, it probably does not by default, ie. requires some active actions by the end user in order to "hack" it, ie. by installing software and perhaps even replacing parts of the bootloader. Ditto for any console really, except other consoles contain various anti-features that the SteamBox lacks related to this functionality.