I don't think the phone can use WiFi to determine your location. That would imply there's already a map of WiFi locations and the phone software can use it. I guess it's possible in places where Google has done it's Street View thing, but to my knowledge the phone doesn't use WiFi for location information.
To answer your question, I've only seen the satellite icon come on when using a program that accesses GPS, not with programs that use towers for location. For example, using Weather Bug, I normally have the "use GPS" option turned off. I don't see the sat icon come on when it updates data or location. But if I set it to use GPS, then the sat icon will come on when updating. So I don't believe the phone uses the GPS radio when using towers to locate.
Now then, A-GPS. Assisted-GPS isn't the same as mobile network. A-GPS (which pretty much all phones use, I believe) is a hybrid. With A-GPS the phone uses the mobile network to approximate your location, and passes this info to the GPS software. By knowing your rough location, and the time, the GPS can lock onto the appropriate GPS satellites much more rapidly than if it had to figure out where it is by searching for all the satellites. Once it's locked onto 3 (minimum) or more satellites, though, it switches to using GPS for location services. A-GPS can also use the tower information to allow navigation if you lose the sat. signal temporarily, because of tall buildings or the like.