I was not having any luck reconfiguring the button maps within the games, because I was not distinguishing, in my mind, the difference between stick configuration in ES, the ES start screen (when you first boot), and MAME configurations themselves. I finally got all the buttons working as intended. Coin credit is on the right side, Player 1 and 2 work correctly, and all buttons are working. I can back out of the games and into ES without issue. All that is left now is to update my ROMS.
ROMS that worked on the original xbox do not uniformally work on the emulators in retropie (examples: DigDug, Defender, Galaga 88, Donkey Kong), but some do. This is easily remedied, though.
Bottom line: everything is good to go. Thanks for your help!
Anyone else looking for help in this issue, include me in the list of people to ask.