The ham is the hind leg of a pig up into the rump
Strictly speaking, what we call "ham" is
cured ham, raw or cooked makes no difference, so yes Americans are cooking a raw cured pig leg joint, the term is the same in both countries
Think of Parma and Serrano, hams eaten cured but raw
A whole raw ham

Bacon is the side part of the pig, mainly the ribs and) is usually cured using a different salt mixture to ham intended for booing,
hence the stronger flavour of bacon; the gammon (again strictly speaking) is the part of the rump that is cured attached to the side of bacon if the ham is not taken off first, and then separated after curing; a bacon joint is any other joint put through the bacon cure I suspect the gammon is removed before curing these days for ease of handling
In practice, I think many Brits don't distinguish between a gammon joint and a bacon joint from the collar, which is my personal preference for cooking at home and serving hot
Gammon

A map of the pig, showing where all the bacon comes from they have decided not to take the ham

So for slow cooker recipes raw ham and gammon are interchangeable, though the ham may be a milder cure