Help! Time Machine old backup not accessible

I have an external 4tb drive with my previous iMac and one large external drive backed up to it (a great system by the way). I now have a new laptop. I used the same 4tb drive, and backed up the laptop onto it. I see the laptop backup plus the old iMac/external backup on that 4tb drive. But I can't access the backups from that old external drive. I click on a file on the external drive, and click "browse time machine backups" (where did "enter time machine" go??). It goes into the stacks-of-windows mode, but there are no backups. At one point a Time Machine window popped up and asked me which backup to use, and I chose the old iMac backup. My new laptop backup is there, those files function normally and can be restored. None of the file names nor external drive name has changed. I've tried restarting. If I manually browse the backup, I can see all the external drive files, but they don't show previous backup windows. I'm afraid to run a new backup of the external drive, because i don't want to erase the months of backup currently on there. And what is the point of the backup if I can't access it from another computer? I'm guessing there is something I need to do, but the Apple support documents are not helpful, and the stakes are high because I don't want to erase that backup, but I definitely want to be able to access it. I will also call Apple Support, but I thought I'd see if anyone else had this issue.