Hi, welcome to the Spine forum!
I figured if I bought Spine, I am free to use my animations created in it in my game with no problems.
Yes, your understanding is correct. The Spine license permits both the use of the Spine editor and the integration of the Spine runtime into your game to implement Spine animations.
The important note is that the Spine Essential and Professional are licensed to a named person. Therefore, you cannot share your license with others.
If you have any further questions, please feel free to ask us.