If I should say the first thing that comes to mind, when I see the question "What kind of multiplayer game do you like?", is "Actually... I don't like multiplayer games". I am almost compleatly only a singleplayer kind of a person.
But if the question is more "What kind of gamer are you when you play multiplayer games", that I can answer easier... because I have played multiplayer games even if I in general dislike them. So, if I go just by that 4 way definition thingy they used in the videos, then I would definatly end up on the right side of things, amongst the Acheivers and Explorers, probably more towards the upper right (Acheivers) more so then anything.
Acheiver is probably the strongest one in me, although I don't really care that much about the "Acheivement" things that most games have these days. It's more of that I love the leveling up thing, be it in actual levels, in skill-level of your skills, weapons, armor, and equipment, etc. It is not uncommon for me to lose a fair amount of interest in a character and/or game once I have reached the max level of my character... which has at times had the effect of that I either stopped playing that character, or stopped playing the game entirely. A lot there depends on how interesting the rest of the game is what decides if I continue playing the character/game to the end or not. That was the reason why it took me about a year and a half befor I had my first max level character in City of Heroes, because I was afraid of that I would lose interest in playing the character once I reached max level with it... and to a degree I did, I still finished all missions etc that I could find, but I still spent less time with those 2 or 3 characters I had which reached lvl 50. End game content has close to never interested me in mmo's, due to the fact that you have to usually play it with a max level character. XD
Explorer would be second for me. Parts of it is definatly that I like to explore... but more so then anything it comes down to that I don't like leaving certain kinds of things unchecked or undone. Be it just something as simple as removing all of the fog of war on a map... or find every single item that is not nailed down and bring it with me, find all quests and hat not, or kill every singel enemy on said map. So it's more of a Acheiver+Explorer combo, with emphesis on the Acheiver part, rather then pure Explorer. I also prefer to finish everything I can find in an area befor I move on to the next.
In third place, pretty far down below Explorer, would Killer probably end up... but not in the sence of attacking and killing other players characters, I have never felt a need to beat my own chest, compare my skill vs others, or assert my dominance over others in that way. I find that to be highly boring, to me PvE is so much more fun and interesting then PvP. XD No, for me this is more the part about trying to get as much ingame cash or what ever out of the things I sell to a market/auctionhouse kind of a thing... so a sort of Acheiver+Killer combo, with Acheiver being the main part. In both City of Heroes and Final Fantasy XI I did spend a fair amount of time tampering with the auctionhouse (done so in most mmo's or multiplayer games where I spent more time then like a week or month). A pretty common thing that I did was buy all the stuff below a certain price which felt reasonably cheap to me, and then turn around and throw the items back into the auctionhouse and sell them at a higher price... but still undercutting players who sold their things at somewhere around a normal price that most players would be ok with buying. I had no problem with getting 5-10% less cash by undercutting the "normal" sale prices, especially not if the items I sold where items I had found as I played the game... since they where essentially free items which did not cost me anything.
The one that I am the least, like pretty far down below Killer, in fourth place, is Socializer. I don't play computer/console games to spend time with others... and no matter who playes those games... be it someone I would consider my best friend of all times, a loved one, or what ever... that can not keep me playing a game once I feel I am done with it, or make me play a game I really don't have an intertest in at all. I play games for the games sake, because I like the game, and because I feel like being alone and all that. This means that first of all I stay away from pretty much all multiplayer stuff in most games which include it. And when it comes to the few mmo's I have played, during roughly a 6 or so year period, I spent the majority of my time in them playing solo. I did join Supergroups in City of Heroes (which was THE mmo for me, played it close to constantly for about 3.5 years befor I finally had played my self sick on mmo's), and I did talk to people there and got a few friends out of it, but I was almost never in the game because of those people. I do play games with friends of mine, but those are close to always things like pen and paper rpg's, tabletop- and/or board-games, and other things like that.
Yeah, I'm mostly an explorer and achiever too. If I'm playing with friends (a.k.a. if the guild that I'm in is a closely-knitted one) I can also rock the social aspect. But, as a player who HATES PvP, I'm definitely not a killer. Of course, in a shooter that's fine, as that's what the game is about. But for MMOs, I literally never do PvP modes. I'd prefer world PvP more (if it's a sandbox game, as that would be realistic, compared to "Battlegrounds"), but it's too much of a hassle I think. You need to be on your toes, be online a long time to be good and to be able to defend yourself/your land, etc. I don't have that time, unfortunately. My real life problems are kinda more important 
