I am having some difficulty understanding your question because you seem to be offering a method of solving a problem (e.g., using mirroring) instead of describing the problem you are trying to solve (e.g., having a redundant system, or keeping a current backup).
Anyway, an easy way to migrate from an iMac to a MacBook Air without having the computer name problem you mentioned would be to use Migration Assistant. This explains:
Regarding Time Machine backup, your question implies that Time Machine storage is somehow equivalent to a computer. Instead, I suggest viewing Time Machine storage as equivalent to a storage device, such as an Apple Time Capsule.
Thank you for your response.
I'm sorry for the confusion but purhaps I am confused and so I don't exactly what to ask. I am going to attempt following the instructions found in the link you left and we'll se how it goes from there.
I was not happy with the way I using the computers so I am going to try a new way.