That’s a UCA202, but you get the idea. In order to eliminate all the delays in the system, you can’t get the headset signal from the computer. You have to get it from the Behringer or whatever you’re using for production. Note that in one case, the microphone itself provides the live production headphone mix.
Anything you get from the computer assuming a “normal” computer is going to be one computer late from internal processing or management. It all takes time. The illustrations in the Overdubbing tutorial are all of devices that can mix the live outgoing music with the returning USB backing track from the computer/Audacity playback for the benefit of the headphones. I call those Perfect Overdubbing because you hear in your headphones exactly what the client or customer is going to hear (given post production, filtering, etc).
Other variations are either going to be very much more complicated (Audacity does not naturally support ASIO), or missing some part of the process – typically, the ability to hear yourself in your headphones.
If you sing in perfect time to the music and your tracks still come out off-time, that can be solved with the one latency that you can change.