Canon's AWB is what Canon calls "Feeling-Based White Balance". It intentionally undercorrects to preserve the general color of the original lighting. It works pretty well for shots made in various kinds of daytime lighting, but most people feel that it is too undercorrected when used with artificial lighting.
The Tungsten preset is for 3200°K professional tungsten photofloods (the same as Tungsten film). This is a considerably higher color temperature than home lighting, which is more like 2700-2900°K.
The Fluorescent preset is for "white fluorescent". There are all kinds of fluorescent lighting with different colors and casts.
Canon's Custom WB is usually very accurate. If you don't want any of the original lighting color left, CWB is the way to go.
My advice is to get a neutral reference—a gray card, WhiBal, or whatever—and take some test shots in different lighting to see how to get the WB on your camera to come out the way that you like it.
Or as everyone else is going to suggest, "just shoot Raw and leave WB until post-processing".