I remember when "Next Gen" meant Playstation, N64 and Saturn. The term seems to refer to the newest generation of consoles that we have any knowledge of. So our current consoles will be "Next Gen" until their successors are revealed in which case those consoles will become "Next Gen".
Bits made it so easy. You had an easy name for a generation of consoles. I guess Nintendo goofed it up by having a 64 bit console competing with 32 bit consoles. What do you call that? 32/64 bit? No wonder the term "Next Gen" was invented. Sony, Nintendo and MS should just stamp "256 bit" on their consoles, regardless of whether it makes sense to or not, and then we have an easy name and we'll all retroactively associate the Dreamcast/PS2/Gamecube/Xbox generation as 128 bit. Then next time around stamp 512 bit on the consoles. All is solved.
Wikipedia has some stuff about first, second, etc. generations but I've never heard anyone use those terms. It's also a little goofy since the Atari 2600 and 5200 are both in the same generation and the first generation consists of stuff like those home PONG systems which aren't really included as videogame consoles. Having the games be seperate from the hardware is usually a important part of what a console is or else stuff like plug-and-play controllers and Game & Watches are all included. With the Atari 2600 being the first machine to fit this description I think if you said "first generation" people would think of that.