[Verdex]

I think I agree with your take on this.

Our brains are already known to lie to themselves to make things appear synchronous that aren't actually synchronous. For example, syncing sound in a TV broadcast makes use of this. And I believe it's the same story with limb motion and tactile feedback. If you touch your finger, then it seems to feel like you touched it immediately despite this feeling requiring some time to work its way to your brain.

One interpretation of the experiment is that your unconscious brain decides to push the button and then tells your conscious brain to make up a story about how it was the one who made the decision.

Another interpretation of the experiment is that your conscious brain decides to push the button, but then your unconscious brain messes with your conscious brain's perception of time such that you'll believe that the button press is happening simultaneously to when you make the decision. The end result being that you "feel" like the slide changes happens before you decide to do it.