Apple ID shows different names
You have detected that the database Apple is using for store reservations is different than the one you see on the My Apple ID page.

You can correct that information by asking them at the bar to assist in correcting and if they say to log in and change it, you can do that on one of the store machines (or your own if you don't trust Safari's private browsing function) and ask them to check that the changes are effective.

Sometimes, you need to call the main AppleCare support number if there is a problem with your record, but most cases you can fix at the time when you see the difference.