SkeletonAnimator vs SkeletonAnimation
These are two different things.
For most Spine runtimes, you use SkeletonAnimation and AnimationState.
When you use SkeletonAnimation, you use the Spine.AnimationState API and that's detailed in the Spine documentation.
SkeletonAnimator was some time ago specifically for people who were very experienced with Unity and strongly preferred sticking the ready-made solution they knew which was Mecanim. If for some reason, you are more comfortable using Unity Mecanim and want to use that, you use SkeletonAnimator. In that case, you would control animations like you would in Mecanim.
If you don't understand Mecanim and its workflow and quirks and bugs, you either need to learn it through Unity tutorials and through normal use, or just use SkeletonAnimation.