Using "act on" like an American

Both of your examples are wrong. The definition says "because you have been given information". So for example:

The police acted on the tip they received.

The manager refused to let me into the store. He said he was acting on the government order that everyone must wear a mask.

It looks like you’re confusing “act on” and the performative sense of “act,” as in, to act in a play.

The performative sense can take an adjective as a direct object to characterize someone’s behavior, as in your first example: act crazy.

The act in “act on” means to take action or to “do something,” as your definition puts it, and the on indicates on what basis (e.g., “because you have been given information”). The phrase isn’t used to characterize the subject’s behavior as having a particular tone. It’s used to indicate why he took action.