Hi,
Thanks for your post in Microsoft Community.
I understand you're having issues with your purchased XBOX showing up as an XBOX peripheral in the device list instead of an XBOX 360 peripheral.
Judging from the current situation, this device is indeed recognized normally at present, but the name is different.
Perhaps the best way to check that the current settings are working is to start a game and see if it works.
If this doesn't work, we can find a solution together.
Please feel free to let me know how it goes.
Kirk | Microsoft Community Support Specialist