It's there, hmm...
Directory of R:\Recovery\WindowsRE
12/22/2012 05:17 AM 201,275,120 winre.wim
I guess it has to do with Push Button Reset. Upgrades operate differently with it. Normally, an upgrade will delete or overwrite the prior version's WinRE.wim. (Well, that's always been my case). Anyway, that is the same date as the Install.wim in partition 6. It delivers the Win 8 recovery environment.
Currently, the WinRE.wim that is in effect is the one in partition 6, because your ReagentC is pointing to it (from page 6)...
PS 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\partition6\Recovery\WindowsRE
Boot Configuration Data (BCD) identifier: 2d25babf-c701-11ea-bc80-f9e613436a82
Recovery image location: \?\GLOBALROOT\device\harddisk0\partition6\RecoveryImage
Recovery image index: 2
Custom image location:
Custom image index: 0
REAGENTC.EXE: Operation Successful.
Shift+Restart will boot to it. I don't know whether that initiates a reset to Win 8, or whether it's just an option that will be offered. And maybe the option is gone, anyway, because the WinRE.wim that was in partition 6 was overwritten by 2004. Yet that same ReagentC seems to be aware there is a recovery image there.
Not sure what else to think.
But on a bravery scale of from 1 to 10, you'd have to be an 8 to try Shift+Restart. If/when there, try "Troubleshoot > Advanced Options > Command Prompt" & report its version at top.