How to recover data from Time Machine

Thanks for the resource listing. As I tried to discover what has happened, looking at my original query, I discovered that the current Time Machine data was migrated not to the new external drive running on 10.11 but instead was put on the old internal drive running on 10.6. Unfortunately, that is a 500 GB drive and now has only 9GB available, which is not going to manage any of the new OS's.

I think the problem occurred because the onscreen instructions kept referring to "This Mac" when it was not clear what Mac they were referring to. Now the desktop that shows up on the old internal drive looks exactly like the desktop on the 10.13 drive that crashed. In turn the desktop on the new external drive running 10.11 looks exactly like the desktop that was on the old internal drive running 10.6. So the OS's on the two drives are exactly where they are supposed to be, but the applications and folders that were paired with these OS's have been reversed. Any ideas on how this situation could be reversed?

Yes, Time Machine is available on the external drive and it is listed in the System Preferences. However, the only backup data accessible on Time Machine runs from Jan 28 to Mar 10 of 2018 despite Time Machine keeping backups since 2015.

To repeat, using Migration Assistant, the complete up-to-date set of applications and files was put on the internal drive instead of the external drive, and the external drive was given the applications and files that had been on the old internal drive. How this could happen is a mystery to me.

If I open Migration Assistant now and identify the Backup drive as the source of the data to be moved, I am asked which Mac do I want the data taken from, the internal iMac drive or the external drive set up running OS 10.11. This makes no sense to me.