With the setups shown in the videos:
In the first video, the piston should extend/be powered like the repeater and redstone dust, but doesn't because it is processed in a different tick ("consumer" tick)
In the second video, the piston is powered, but the repeater is not, and the dust is VISUALLY not powered, because they are processed in a different tick ("producer" tick)
In both situations, the wire, repeater and piston should, logically, all be powered.
This behavior is not only inconsistent with java, it is also very unintuitive. It makes no sense to behave this way.
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After studying this report for a while and comparing all of the comments on MCPE-15793, I've concluded that it is the same issue. I will copy over the videos and description to that ticket.