(1) Here are your two problems as I see it...
(a) Right, it is hoped by deleting Partition 5 & recreating it, Acronis will no longer find bad sectors in it. The recreated partition will occupy the same sectors as the deleted one had occupied.
I can't tell whether (a) was solved. But your Partition 5 is currently not used at all. If you don't want to transform it into the Active recovery partition, then just delete it with Partition Magic & bring it back as a regular partition (not a recovery
partition). Then see whether Acronis likes it. But remember always to reboot after doing partition work like that.
(b) It just can't be right that each fit & proper system image you make of Win 10 must include that dinosaur of a Gateway Win 8 in it. The way to prevent that is to make Partition 5 your active Recovery partition.
What I did definitely solves (b). See how Windows system image backup now requires my new Recovery Tools (X:) to be in the image, but the old WinRE 2004 (R:) is offered as an optional...

It would require Partition 5 for you & offer Partition 6 as optional. You need make just one image backup of Partition 6 because it never will change from 2012. But Partition 5 will change (at least) whenever an upgrade/repair-install is done.
But a peculiar side effect is that your Partition 6 will no longer have a WinRE.wim in it. I think all else will remain. Nothing in my instructions wipes it away, but it goes. Also, Partition 5 will have a Recycle Bin that apparently will hold the recovery
files we XCOPY into the partition. But that is harmless & could disappear on its own in time. It suggests something causes Windows, itself, to put recovery files into Partition 5 & throw ours into the Bin.
[Edit: I no longer think my Xcopied files were put into the Recycle Bin, but possibly they did get over-written with a new set provided by ReagentC.]
Another peculiarity is that it took two tries with Shift+Restart before the new Recovery partition worked well. But I think that's due to I forgot to reboot after my partition work in DiskPart before doing ReagentC to point to the new one. And/or I should
have booted once to Windows before trying Shift+Restart.
I'll post the ReagentC procedure that does all that below.
(2) "I’m a bit confused. Is it finally called part 5 and in part 5 location?"
Yes, but best do a reboot after partition work like that to be sure the system really knows it. I've been meaning to bring this up, myself. The instructions were...
Select Part 5
Delete Partition
Create Partition Primary
I think it might be recreating the partition into the newly created unallocated space (from the Delete command) because the focus is still on it from the Select command. But possibly/probably it puts it back into that space because there is no other unallocated
space on your HDD (or mine) that precedes it. If there were, we'd have to consider an additional parameter/two in that Create command, like giving it a Size or Offset...
SIZE=<N> The size of the partition in megabytes (MB). If no size is
given, the partition continues until there is no more
unallocated space in the current region.
OFFSET=<N> The offset, in kilobytes (KB), at which the partition is created.
If no offset is given, the partition is placed in the first disk
extent that is large enough to hold it.
OMG, who wants to go figure KB's? So, we are lucky to have only the one span of unallocated space & just the right size (after our Delete). Or, best use a 3rd-party tool for it, & just do the SET ID in DiskPart.
(3) "So, you created a new part. Then you deleted and recreated it? Why? Then you did something to make it a Recovery part."
I did it just to see whether it would work to possibly solve the two problems you've got. But also I learned from it & might learn more when I undo it. And DiskPart is part of Windows.
(4) "The copy X (part 6) Recovery to R (art 5) Recovery. So copying everything from Part 6 to 5? Or, how does it know to take only the wim?"
XCOPY X:\Recovery R:\Recovery /E /I /H /K /X
That copies the Recovery folder & all it contents & subfolders from partition X: to R:. That includes the WinRE.wim. Nothing else is copied. But I believe something caused Windows to put all those files into the Recycle
Bin it created on R:\ & replace them with its own set. I'm going to experiment with ReagentC to see whether it will put a set a files back into my original partition w/o having to do an XCOPY.
(5) "Not clear, did you work out how to get it right?"
Definitely, I'll get my files back into my original partition one way or another. And ReagentC will point to it. Maybe I'll delete both partitions & try a repair-install.
(6) Here is the ReagentC procedure that I used to redirect the location of RE from my Partition 5 to the new Partition 4. But it won't do it in reverse because it can't find the Path to partition5 \Recovery\WindowsRE which
no longer exists as a surprise to me. The Recovery folder is there, but not the other.
C:\WINDOWS\system32>ReagentC /Info
Windows Recovery Environment (Windows RE) and system reset configuration
Information:
Windows RE status: Enabled
Windows RE location: \?\GLOBALROOT\device\harddisk0*partition5*\Recovery\WindowsRE
Boot Configuration Data (BCD) identifier: d1890737-c25c-11ea-b1a7-74e6e2074932
Recovery image location:
Recovery image index: 0
Custom image location:
Custom image index: 0
REAGENTC.EXE: Operation Successful.
C:\WINDOWS\system32>ReagentC /Disable
REAGENTC.EXE: Operation Successful.
C:\WINDOWS\system32>ReagentC /Info
Windows Recovery Environment (Windows RE) and system reset configuration
Information:
Windows RE status: Disabled
Windows RE location:
Boot Configuration Data (BCD) identifier: 00000000-0000-0000-0000-000000000000
Recovery image location:
Recovery image index: 0
Custom image location:
Custom image index: 0
REAGENTC.EXE: Operation Successful.
C:\WINDOWS\system32>DiskPart
Microsoft DiskPart version 10.0.19041.1
Copyright (C) Microsoft Corporation.
On computer: PCRSCOMPUTER
DISKPART> Select Disk 0
Disk 0 is now the selected disk.
DISKPART> List Part
Partition ### Type Size Offset
------------- ---------------- ------- -------
Partition 1 System 99 MB 32 KB
Partition 2 Reserved 16 MB 100 MB
Partition 3 Primary 929 GB 116 MB
Partition 4 Recovery 827 MB 929 GB
Partition 5 Recovery 847 MB 930 GB
DISKPART> Select Part 4
Partition 4 is now the selected partition.
DISKPART> Assign Letter=R
DiskPart successfully assigned the drive letter or mount point.
DISKPART> List Vol
Volume ### Ltr Label Fs Type Size Status Info
---------- --- ----------- ----- ---------- ------- --------- --------
Volume 0 D DVD-ROM 0 B No Media
Volume 1 C PCR Hard NTFS Partition 929 GB Healthy Boot
Volume 2 R Recovery To NTFS Partition 827 MB Healthy
Volume 3 WinRE 2004 NTFS Partition 847 MB Healthy
Volume 4 ESP FAT32 Partition 99 MB Healthy System
DISKPART> Exit
Leaving DiskPart...
C:\WINDOWS\system32>ReagentC /SetREImage /Path R:\Recovery\WindowsRE /Target C:\Windows
Directory set to: \?\GLOBALROOT\device\harddisk0*partition4*\Recovery\WindowsRE
REAGENTC.EXE: Operation Successful.
C:\WINDOWS\system32>ReagentC /Enable
REAGENTC.EXE: Operation Successful.
C:\WINDOWS\system32>ReagentC /Info
Windows Recovery Environment (Windows RE) and system reset configuration Information:
Windows RE status: Enabled
Windows RE location: \?\GLOBALROOT\device\harddisk0*partition4*\Recovery\WindowsRE
Boot Configuration Data (BCD) identifier: de89953b-cd80-11ea-b1b2-74e6e2074932
Recovery image location:
Recovery image index: 0
Custom image location:
Custom image index: 0
REAGENTC.EXE: Operation Successful.
C:\WINDOWS\system32>DiskPart
Microsoft DiskPart version 10.0.19041.1
Copyright (C) Microsoft Corporation.
On computer: PCRSCOMPUTER
DISKPART> Select Disk 0
Disk 0 is now the selected disk.
DISKPART> List Part
Partition ### Type Size Offset
------------- ---------------- ------- -------
Partition 1 System 99 MB 32 KB
Partition 2 Reserved 16 MB 100 MB
Partition 3 Primary 929 GB 116 MB
Partition 4 Recovery 827 MB 929 GB
Partition 5 Recovery 847 MB 930 GB
DISKPART> Select Part 4
Partition 4 is now the selected partition.
DISKPART> Remove Letter=R
DiskPart successfully removed the drive letter or mount point.
DISKPART> Exit
Leaving DiskPart...
Next time I'll do a reboot to Windows here -- before trying Shift+Restart.