Dead on again.
What this and your other comment make clear is that once you start talking about the probability that X is true, especially in the context of hypothesis testing, you've moved (usually unwittingly) into a Bayesian framing, and you better make your priors explicit.
You throw a coin five times and I predict the result correctly each time.
#1 You say that I have precognition powers, because the probability that I don’t is less than 5%
#2 You say that I have precognition powers, because if I didn’t the probability that I would have got the outcomes right is less than 5%
#2 is a bad logical conclusion but it’s based on the right interpretation (while #1 is completely wrong): it’s more likely that I was lucky because precognition is very implausible to start with.