I beg to differ. 240V household power is ONE phase. It most certainly does not use "multiuple power sources." The power source is a single leg of what was almost assuredly a three phase supply. The transformer near your house drops it down to 240V (single phase) with a center tap that provides the 120V that most of your household stuff use.
When you come to discuss motors, you have to talk about what sort of motor you are talking about. Smaller motors (universal or the like) are usually single phase devices that could't use three phase even if it were available. Larger motors are typically induction motors which ,can be three phase for practical purposes or single phase (though the latter requires special provisions to get them started). Most of the commercial stuff comes three phase. If you have three phase available to you, then it would be handy to have it in the shop. Otherwise, you have to deal with single phase motors or deal with a couple of ways of getting a three phase motor to work with a single phase feed.
Wire size doesn't have much to do with phases. The higher the current, the larger the wire size needs to be (and the more loss there is). Since power is voltage times current, you can get by with less current at higher voltages. This is why your motor will run on 9A at 240 but needs 20A at 120.
The other neat feature is that a lot of variable speed stuff these days (like some lathes, etc...) are essentially three phase motors but the phase (and the frequency) is electronically generated from a single phase source (essentially DC).
I've been meaning to write a more concise description of all this. You may see this as an article soon.