franz1 wrote:
. . .
The top one has icons showing the two partitions from the external drive and the Air's HD.
That sounds like the Today (Now) display, the one that's show in full when you first Enter Time Machine. It's not a backup; it's the current contents of your Mac.
When I clicked on the 2nd window, it seems to be a back up from July 20, which I remember making on the old MacBook. It has one folder on it with my username.
Yes, that's a backup. 😉
When I click on this folder, I find 4 folders inside it titled: Applications, Users, Library and System. I clicked on Applications and it was empty
That sounds like you'd excluded Applications from backups. Or else you didn't wait long enough for all the apps to be displayed.
sometimes the names of programs covered with a see-through "no parking" sign - a circle with a slash through it
That's an incompatible app.
Is this the Time Machine's records?
Yup.
Can I click on the July 20 date and click on restore? Will everything just come back on it's own?
No. You can restore individual items that way (if you have permission to them), but not a whole system. You can restore "simple" apps that way, too, but not "complex" ones -- ones that came with their own installers, because they install other files in other places. If you don't restore those other things too, the app won't work well, if at all.
Can you post a screenshot of the Migration Assistant window that should show your backups (something like this):