Actually you can win without an advantage (e.g. fools mate, legal mate, etc) but it wouldn't be wise to always try winning that way. Or as just stated: a king up! Lolz.

A King up..!

From my own experience studying endgames srn347, i have found that you don't need that much advantage in order to win, however, you need some kind of advantage.  You cannot expect to win without something to exploit.  The best you can do is play for the draw, if that is the case.  If you lack advantage, then you must seek to prevent your opponent from materializing any advantage of his own. 

My question asks for a generalization, for example, wikipedia says 5 except when pawns are involved.

You really answered your own question SRN -- there isn't a right or wrong answer, in fact, there isn't even a general rule. You can be behind in the end game and win easily enough if you have better positioning than your opponent, or you can be ahead and finish the game without even having to worry about pawn-promotion. Every game is different in that regard -- often times games of chess start out the same, but they never end alike.

Htat's why I say in general instead of asking for one specific right answer.

I think it all depends on the situation and the piece power you have.