Time machine wants to inherit the last backup I made on the same computer

The dialog is asking if you want to inherit them, meaning do you want that Mac to use those earlier backups as its own. If you choose to do that, they will no longer be available to the Mac that created them.

I see this answer a lot, but it's a bit vague. Does "no longer... available to the Mac that created them" mean that the Mac that created the backup can no longer READ the files? Or only that it cannot write? I suspect the latter - but there's no telling with Time Machine.

Personally, I've found Time Machine to be untrustworthy. I had a recent crash but thought I was okay because I'd done nightly backups. When I restored the system to another machine, I found files missing - the most critical being one of my keychain files. Imagine suddenly no longer having over 50 passwords to various accounts! I was able to recover the missing keychain by scrubbing the crashed disk - but am a bit stunned that Time Machine failed me in such a non-obvious manner. Other files are missing too - and no, they were not excluded by the Time Machine options dialog.

Given the intended function, such failure is inexplicable. It means Time Machine missed that file dozens - if not hundreds - of times.

Just click on the pull down menu and select the correct computer.

….earlier versions were but not the last one

The dialog is asking if you want to inherit them, meaning do you want that Mac to use those earlier backups as its own. If you choose to do that, they will no longer be available to the Mac that created them. The choice is yours to make.

Alternately, if you hold an option key while selecting the TM icon, the option to Browse Other Backup Disks will appear.

When I open time machine and click on the lock. I type in password and then turn on time machine. I automatically get a pull down menu to select from.

If the drive was used by another Mac for Time Machine backup, when you first connect the drive to the Mac and select it as a Time Machine drive, a dialog will appear with the information that it has been used by another Mac. You can then select which backup you want to use.