On the first 2 screenshots : the circuit
On the last :
1) Expected output in the chest when you press the button
2) Output obtained after pressing the button
3) Output obtained after pressing the button after a logout/login. Which is incorrect ! 😞
I think it comes from hoppers that refresh too fast or too slow compared to the others. This behavior happen often and is very random (seems to depend of the XYZ position of the hopper).
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This hopelessly breaks so many different hopper functions, that any kind of synchronization or timing becomes an issue. When using hopper systems, no matter how well timed all of the other components of the system are, the whole thing becomes dysfunctional the second you log out. Very frustrating and took me forever to find! Please please please fix this Mojang. Pretty please?
Thanks a lot for reporting this so clearly, Marc!

Is this still a concern in the current Minecraft version 1.8.1 Prerelease 3 or later? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.

This is certainly still a concern in 1.8-Pre3. The test is extremely easy to do.
I tried it and got the wrong ordering before I even logged out. After re-logging, it was even worse.
Changing the location of the power source changes the outcome further.
It's possible that this is related to MC-11193.

Reopened, thanks

Is this still an issue in the current Minecraft Snapshot 15w47c or later? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.

Ticket resolved as incomplete, because no answer in a reasonable amount of time (1+ year), if it still happens, please update the ticket.