This is an interesting topic, and one that is obviously near and dear to a lot of hearts here. After all, as a group, we have a lot more interest in the ins and outs of multiplayer than, say, a forum of match-3 fanatics.
I don't have a ton of interest in copying and pasting stuff, but I'll add some thoughts on the categories. First, I think notifications and app badge should just be combined; they are essentially the same thing. As long as a game can communicate the fact that it is your turn, does it really matter how the information gets there? Plus, the Notifications Center in iOS really is pretty good and improves with every major update (especially considering how bad it used to be).
Next, I have two more categories that are pretty important to me ... honestly, I'm kind of surprised they aren't mentioned. First is starting a game. You'd think it would always be easy, but that's not the case. The poster child for this in Neuroshima Hex — a game that should have about 10 active threads here, yet has none. Why? Because no one has half an hour to try to get a game started.
A couple of years ago, over at That Site That Used To Be Good, I posted the results of a casual experiment comparing how easy it was to start a game in two popular strategy-game apps. The results:
Lost Cities: one step to send an invite
N. Hex: THIRTEEN steps to send an invite.
I'm still shocked by that. Putting up such a barrier to online multiplayer games is a huge designer fail — and, of course, the No. 1 reason that N. Hex has just withered away and become irrelevant. It's a great game and an otherwise solid app ... but only the truly hardcore have the patience to attempt a multiplayer game.
A second missing category is player base. Is anyone actually playing the game in question, or does the matchmaking lobby have tumbleweeds blowing through it? Is even trying to get a game going an exercise in heartache?
Sadly, there's too many examples of this, and I'm sure we've all got some personal disappointments. For me, Kahuna will serve as an example. There's a game that technically has online multiplayer ... but I have not heard of anyone actually completing a game in at least a couple of years. The developers might as well have skipped it as a feature.
Anyway, that's my 2 cents, FWIW. Like I said earlier, this is a great idea for a topic. I'm hoping it gets lots more responses.