Two Time Machine backups on same disk

Dear apple community, for my MacBook I regularly make backups on several external disks. This summer I bought a new MacBook and used the same external disks for backups. I was asked whether I want to start a new set of backups or "claim" the old one. I chose a new one, assuming this will start independent backups. This seems to work well for individual files (except for a confusing message when backups are made), but now I am in need to restore my OS system. When I boot so I can see the old versions on the external disk, I am offered options to restore the system from time machine, but only the versions of the old computer (not the one of last week that I need; the old computer had a much older OS). If I had a fresh and complete backup, the new system should be there and accessible -- shouldn't it? How can I tell the system to find it?

Not sure whether it is related: here is the confusing message. When I ask for a fresh backup, Time Machine does what it should, informing me about progress the usual way. Only when it's finished, it says "last backup completed" was long ago, ignoring all new backups and referring only to the ones for the old computer. Of course I checked, and all my files are stored on the backup as they should be. Is this a hint that the "new" set of backups isn't quite as complete as it should be?

Any suggestions on how I could access a backup of my OS as of last week?

Thanks a lot!