Examples of verbs that don’t describe an action or state of being

You’re right, my error. I think what the article is describing is linking verbs, which have a different definition to state of being or doing verbs.

That’s what I’m thinking, anyway.

The ‘doing’ verb there is ‘to think’ and the linking verb is ‘to be’. Some websites will say ‘to be’ in that example is a ‘state of being’ verb, my guess is the author is stating that’s not the case with Merriam-Webster.