(1) I am overjoyed all is well thus far. It all looks great to me. Good that you did the reboot afterward. I wasn't sure, either, Acronis would be satisfied. It was just a hope that delete/recreate would go deep enough into the works to make a difference. Must be it woke a sleeping giant in the HDD firmware to set things right. It was when you got to the SET ID command that the partition became a recovery partition. But as yet it's empty.
(2) Let's go ahead & put the Recovery folder into it. XCOPY will definitely not remove files from Partition 6. It will just make a copy into Partition 5.
DiskPart <<< Enter DiskPart
Select Disk 0 <<< Select the HDD
List Part <<< Show the partitions
Select Part 5 <<< Select the new & improved Recovery partition
Assign Letter=R <<< Give it letter R
Select Part 6 <<< Select that Push Button Partition
Assign Letter=X <<< Give it letter X
List Vol <<< Show the letters
Exit <<< Exit DiskPart
XCOPY X:\Recovery R:\Recovery /E /I /H /R /K /X <<< Copy Recovery folder from Partition 6 to 5
DIR R:\ /a /s <<< Show what's on Partition 5
DiskPart
Select Disk 0
Select Part 5
Remove Letter=R
Select Part 6
Remove Letter=X
Exit
(3) Let's make Partition 5 to be the active Recovery partition. But /Disable will delete the contents of the Recovery folder on Partition 6. So, we'll copy it back afterward...
DiskPart
Select Disk 0
Select Part 5
Assign Letter=R <<< Put R: back onto Partition 5
Select Part 6
Assign Letter=X <<< Put X: on Partition 6
Exit
ReagentC /Info <<< Shows Partition 6 is both the RE location & the RE image location
ReagentC /Disable <<< Turns off the RE location, moves WinRE.wim out of 6
ReagentC /SetREImage /Path R:\Recovery\WindowsRE /Target C:\Windows <<< Set RE loc. to P. 5
ReagentC /Enable <<< Turn it back on
ReagentC /Info <<< Shows P. 5 is the RE location & P. 6 remains the recovery image loc.
XCOPY R:\Recovery X:\Recovery /E /I /H /R /K /X <<< Copy Recovery folder from Partition 5 to 6
DIR X:\Recovery /a /s <<< Show Partition 6's Recovery folder
DiskPart
Select Disk 0
Select Part 5
Remove Letter=R <<< Remove letter from Partition 5
Select Part 6
Remove Letter=X
Exit
That should do it. We'll discuss Partition 6 afterward. At least it is out of the way of a Windows system image now. Can you use Acronis to backup files instead of disks or partitions? Or we can use XCOPY to copy all of that push button partition's files somewhere safe. Then we can delete/recreate it & copy them back -- pretty much like we've already done with just the Recovery folder.
To see everything that is on Partition 6, put letter X on it, then "DIR X:\ /a /s" should do it. Post it, if not prohibitively massive.