Here are histograms of the longest runs of consecutive heads in the sequences you generated. How do the imagined sequences compare with the real ones?
When I attempted to identify the coin toss sequences you generated, I looked just at the longest run of consecutive heads. If there were 5 or fewer heads in the sequence, I declared the sequence to be a fake (imagined tossing). If there were 6 or more, I declared it to be real (actual tossing).
Redrawn below are histograms of the longest head runs for your imagined and actual sequences. I colored in green the outcomes I thought were real and colored in red the ones I thought were fake. In a perfect classification the top ones would all be red and the bottom ones all green.
