From what I understand, it is only RAW potatoes that are bad. Mine have eaten cooked regular and sweet potatoes with no harm done. In fact, this morning they even had a few left over tater tots with their breakfast.
I don't think even raw potatoes are toxic if they don't eat a pound of them, though they'd likely have a tummy ache. Mine eat leftover cooked in the skin taters all the time. They really love fingerlings.
Green - just like for people . . . and cooking doesn't neutralize the solanine in green potatoes. The potato sprouts are bad, too. Finally, it isn't the green itself but that's the indicator of the solanine.