Does stock image have to match device?

So I purchased a used Nexus 6 on Swappa that came with an unlocked bootloader and a custom ROM... Just so I can start totally clean (and party my OCD), I want to flash a stock image, relock the bootloader, and then do any OTA updates I may need... The problem is that since it's a custom ROM it doesn't tell me inside "About Phone" which device I have... When I go to google's site for the stock images it has several carries listed etc...

My question is, does it matter which stock image I flash? I'm T-Mobile.. so if I flash a stock T-Mobile image, will that matter if the device is not from T-Mobile? The other thing that's throwing me off is there's two stock T-Mobile 5.1.1 images on google's site, with different device ID's... so I have a feeling it does matter which version you flash.

Thanks!