~snip~
Glad that it helps as it should be, thank you for your words.
Yes, you got it right. I've already used this copy of Google's two-factor authentication several times when I transferred it to a new smartphone. There was also a case when I had to reset the data to factory settings and, accordingly, without a copy of my keys, I would not be able to restore access without the help of the support service.
Great info, I didn't know that. When I get my new phone (still with a back up one), I will try it as well.