What is the best way to preserve my last computer backup?

I got a new MacBook Pro a month or so ago. I want to save the last backup of my old MacBook Pro just in case.

I set up the new computer to save to the same external drive that the old computer used. It took me some time to notice that when plugged in, the hard drive is showing up as two different drives. I can eject each one separately, but I cannot access the backups of both computers on either one of the drives. I also can't create a new folder on either of these drives which I don't believe is how it used to work.

I don't remember exactly how the original Time Machine setup went but it must have made me create "Backups of MacBook Pro (2)" which I thought would be a separate folder on the same external drive. Instead, it created a completely separate external drive.

I've been getting error messages that the backups aren't happening (even though the disk drive for "Backups of MacBook Pro (2)" [new computer] shows back-ups for the last week.

Today I'm trying to fix it, with the first step Time Machine gave me this message:

I chose "Start Separate Backups" because I don't want to erase the backup of the old computer. It gave me this error message:

But my computer is showing that the disk is connected.

I'm pretty sure all these error messages are coming because it's not reading the backup to the previous computer correctly.

If I "Claim Existing Backups" will it get rid of the old backup when the external drive fills up? Will it eliminate the two separate drives and stop these annoying error messages? Is there a better way to preserve the backup of the old computer?

Thanks in advance for any help provided.

[Re-Titled by Moderator]