AT&T shows it as a Motorola MX12405445 because that's how the device appears in the official international IMEI database.
Even though any US Nexus 6 is an XT1103, Motorola still has to have a way to classify them internally.
XP12357045 are devices whose IMEI numbers follow normal GSM Association standards, meaning any phone sold by a GSM carrier or sold SIM-free through general retail channels like Motorola or Google Play. MX12405445 are devices that were sold by the US CDMA carries, meaning Sprint, Verizon, or USCC.
To make it simple, think of it this way. If the XT1103 device's IMEI number starts with 99, it's MX12405445. Otherwise, it's XP12357045. Simple as that.
As for who handles updates, it all comes from Google, no matter what. However, Google will withhold updates based on SIM card pending carrier approval. AT&T has not approved Android 5.1 yet, and therefore, anyone with an AT&T SIM card in their Nexus 6, no matter where it was purchased, will not be offered the OTA until AT&T approves it. So far, the only US carriers to approve Android 5.1 are T-Mobile and Sprint.
If you have the original Sprint SIM card that came with the phone, put that in your phone and give it a few minutes. You'll most likely be offered the OTA to 5.1 / LMY47D.