Indexical Banana

There's either a bug in your script or the original paragraph is really mundane. You're not counting the number of occurrences of a given sequence, you're counting the number of 20-flip sequences that include it. An string of HHHHHHHHHHHT will just count as one for each.

Obviously, you can have HHHT without getting HHHH. But if you get HHHH, then you still have the HHH prefix, and now you have another flip to get T (and if that's H, yet another chance). The only way you get HHHH without HHHT is if after 3 Hs, it's Hs all the way to the end.

There is a subtle effect where, given that you're seeing a finite n number of flips, it is more likely that those n flips contains more HHHT than HHHH than vice versa (balanced out by a greater chance of larger leads for HHHH). But that's much smaller and subtler than your pic.

And "does take less time to show up" doesn't make any sense to me, if it means which occurs first. Neither occurs until you get HHH, and then whichever flip comes next is 50%. My guess is a reporter credulously paraphrasing without understanding. (But maybe I'm missing something)

Your explanations are exactly correct. The simulation is correct for the phenomenon the OP describes, but the concept is rather contrived and poorly explained, and isn't an interesting result for any practical purposes I can think of (except hustling foolish bettors).

see https://www.nowpublishers.com/article/OpenAccessDownload/RBE-0092 …

This is a poor example for the bias bias phenomenon Gigerenzer is trying to illustrate. This is a case of not parsing and understanding the problem correctly, not a case of having perceived experience tell us that HHHT is more likely than HHHH.