WHAT IS AP?
In case you aren't aware what AP means, I suppose I should try to define it. AP stands for Analysis Paralysis. What it means is that a player can get stuck analyzing a situation on their turn for so long that they seem to be paralyzed with indecision and, in fact, they likely are - they can't get to the point of making a decision and completing their turn.
But, there's an important distinction here for what is expected/necessary for a game. Chess typically has players competing 'in their minds'. They are thinking of hundreds of possible move combinations in their heads. They are planning out multiple moves in advance for themselves while also trying to anticipate what the other player is trying to do and prevent anything catastrophic. This is especially important when you consider that money or prestige may be on the line. This isn't to say you can't plan out several moves in advance in other games, just that it's more critical in chess with the idea that several lines of planning should really occur to be successful.
However, I don't know that I would define that sort of behavior as AP. The reason is because they are still DOING something in their mind - it just takes a long time. But, it's still possible for AP to creep in, to get to the point where the analysis is leading nowhere and a decision cannot be made. This is why they have chess clocks, to force players to manage their time and keep them from falling into an AP mode.
On the other hand, AP gamers take a longer than expected amount of time on their turn. This idea is what I think is the crux of the problem - the EXPECTATION of HOW LONG a turn should actually be during a game. If one person is taking longer on their turn than everyone else expects them to be taking it, then they might be considered an AP player by that group, even if they aren't 'spinning their wheels' so to speak.
The interesting thing with this idea is that AP may mean different things to different groups. I, as an AP player, love taking my time on a turn and not feeling rushed about it. With some people I play with, me taking more time on my turn is fine and they aren't bothered by it at all. Other times, I might be playing the same game with someone else and, if I don't take my turn relatively quickly it becomes a problem for them.
How do I know it's a problem? Well, I'll get to that in a bit.