You are welcome.
(1) A directory or folder is a container that holds files. Install.wim is just one file (a pretty big one) among many that are on Partition 6. Do you want the whole partition or just the file? If you want the whole partition, you should keep the Recovery folder too. It was there to begin with, but 2017 is the last year in which it had its original Gateway contents (a reworked WinRE.wim). That is why I say Partition 6 is dysfunctional. The reworked WinRE.wim had your push button in it, I think.
If you want the whole partition, don't do it until we adjust the XCOPY commands. Do you want it?
(2) GatewayWin8 is a folder that will temporarily hold the file Install.wim while we delete/recreate Partition 6. But that will need to be adjusted, if you want the whole partition.
(3) "Will moving it to the external drive make it visible"
I think it should become visible because the external drive is not a Recovery partition, But there might be something in the file's "permissions' that will keep it invisible in File Explorer. But DIR (with /a) in a Command Prompt should see it just fine. Also, turn OFF "Hide protected..." in Folder Options (type into search), & you should see it in File Explorer too...
You still won't be able to delete it in File Explorer without a huge battle with permissions (that we likely will lose). It even may refuse to delete in an Administrator Command Prompt while in Windows. In that case, we'll have to go to Advanced Startup & do it there.
(4) "get the number on Part 6 correct"
The number you put there is the partition you will delete when you get to "Delete Partition Override". Examine the "List Part" just above to be sure it is the Push Button partition. It should be. If so, don't inadvertently type in a different number than 6.
(5) Well, to get the whole partition here are the XCOPY commands (using letter E:)...
XCOPY X:\ E:\GatewayWin8 /E /I /H <<< Copy old Partition 6 to E:\GatewayWin8 folder
XCOPY E:\GatewayWin8 X:\ /E /I /H <<< Copy it back to the new Partition 6
Also, for that second one, omit the MD command which was: "MD X:\RecoveryImage".