Giant_Joe - maybe I’m being pedantic. I do not doubt that you understand it. But I was just pointing out your terminology is not consistent with Mises, that’s all. Your last sentence is a tautology. Action is defined as purposeful behavior. So saying “man acts purposefully” is saying nothing except “man behaves purposefully purposefully”.
The axiom does not say that all behavior is purposive. This is clearly not true. But SOME behaviour is purposive. To deny the action axiom, would itself be an action. So in the act of someone asserting that “man does not act”, they would be refuting their own statement. That is what a self-evident truth means.
I asked a similar question myself a few months ago: here.