If a verification e-mail is being sent to a me.com address, it's because you have a me.com e-mail address associated with the Apple ID that Apple is trying to verify. When you set up a Mac, you fill in some boxes, and an Apple ID is set up automatically. I suspect that's the problem.
You will probably have to log in to each Apple ID and check the e-mail address associated with it. That may or may not be the Apple ID name.
If you find an ID with a me.com e-mail address as the name, you can change the contact e-mail address. You can also set up a new me.com address in mail and check that.
Discussion of issues related to iCloud, Apple's "cloud" service designed to store and sync your music, photos, apps, calendars, and documents
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