Team-based stuff is definitely tricky to classify. Is it co-op or is it PvP? There are attributes of both in there, and different people treat it different ways. For me, I see it through the lens of co-op, since I can play with my wife and thus we can be doing something together, rather than one of us just watching the other play (which is lame and boring). And I don't game on my own anymore.
For others, perhaps the competitive aspects are more important, and that's great. But honestly, the amount of time I have for gaming means that I'm never going to be putting more than 70-100 hours into most games, and even that's pretty rare. For the average FPS game, if it's a 10-20 hour solo/co-op experience I'm satisfied with that, as usually any more would just be padding out the experience. There are always higher difficulties I can play on, or I can come back in a year or two when it's feeling more fresh again.
Playing with actual human opponents is both a blessing and a curse. At advanced play, it can indeed be more interesting. At lower levels, it can be extremely off-putting because there's no real difficulty controls. Sure there is supposed gating to keep advanced players out in a lot of games, but it only works so-so.
Warhawk is a good example of a game with this really exciting premise -- FPS with the ability to get into planes! -- but when you play with real opponents you wind up only being able to survive in the air for 10-60 seconds max, it seems. That's... something of a letdown. It's not exactly the Top Gun scenario most people picture, I don't think. For my wife, it was so intensely frustrating that she just stayed on foot or in ground vehicles, but even that could get really frustrating.
Contrasted with us playing Unreal Tournament or Counter-Strike, we were able to play those against bots just for hours and hours and hours. The bots would occasionally do something surprising, which added some spice to the affair, but largely we existed in a quasi-predictable space where we could test ourselves and see what interesting things we could do, and work on our skills, etc. And that was the whole game for us. To us, UT is as much a co-op game as L4D is, as that's the only way we play it. To others, UT is the very definition of a PvP game.
The problem comes when, as has been happening lately, the bots are left out. Then it's just PvP, unequivocally.