@einstein1969

Is the partition 4 a manufacturer partition with an "out of the box" image to restore the initial Windows should it be needed, or is it just made and controlled by you? In the latter case just ignore this. In the former case I'm not expert but it could make some things different.

In the part I, when you say "I therefore preferred to make room with Minitool Partition Wizard 12.7. And I added about 300MB to partition number 5", I think that this somehow disabled WinRE (*), caused the "RAW" problem in partition 5, and WinRE decided that partition 6 was its. I have been told about this "RAW" thing as a "bug" that PW can cause, although I used PW in a "standard configuration" (MBR disk with 3 partitions; System Reserved 50MB, Windows C: 223GB, Recovery Partition 522MB -> 2GB) and I didn't experience it (although I couldn't install KB5034441 either for the same 0x80070643 error).

(*) When the user runs ReAgentC /disable , it not only does the disable but it also moves Winre.wim from the recovery partition to C:\Windows\System32\Recovery (ReAgentC /enable does the opposite). This allows even deleting the Recovery Partition temporarily, as MS does in its provided directions in your second link. But as something disabled your WinRE in a buggy way, your Winre.wim was lost and ReAgentC /enable didn't find it, until you could recover it with reagentc /setreimage .

Topic: KB5034441 — Windows Recovery Environment (WinRE) successful install @ AskWoody

[4] I doubt that Winre.wim is old. Can you create problems using an old winRE? If yes, how can I update it?

"Problems" or not would depend on if you use BitLocker among other things. The link above provides a method to install KB5034441 that precisely moves WinRE to C:, applies KB5034441, and moves WinRE back to a Recovery partition. You would be now in the 2nd said step if you want to try, but read below.

[3] I would like to move the winREr under the restoration partition. How can I do?

To just change the primary partition back into a recovery partition use set id=27 (or for GPT set id=de94bba4-06d1-4d40-a16a-bfd50179d6ac) You should use the GPT command (plus gpt attributes =0x8000000000000001), but you have at least three problems: the 9 GB partition in the middle, you've already restored WinRE explicitly to C:\ with a Recovery partition set, and your MiniTool Partition Wizard seems to have caused you problems (btw the latter version isn't 12.7 but 12.8).

If at all, I would delete the "Recovery" partition first with diskpart (it's being read as "Recovery" because it has the ID, but your WinRE is enabled in C: instead), exit diskpart, possibly reboot, re-enter diskpart, create a 1 GB partition between C: and the 9 GB one (so you get rid from your non standard configuration), and set it as Recovery with set id=de94bba4-06d1-4d40-a16a-bfd50179d6ac and gpt attributes =0x8000000000000001 .