Finally figured out what is going on.
There is a sentence in the documentation of Overdubbing I had missed.
To the effect
".. make sure you are able to hear external input while you are playing some track in Audacity.."
This precondition was not met in the Laptop I was using.
I checked this on a Third Laptop (bit older than the other two) and this condition was met and the issue I was facing vanished.
I am able to listen to tracks while recording a new track. The new track does not have the audio of other tracks.
Hope this is useful for someone else who may run into this issue.
Appreciate your help and your strong insistence that it should work.
P.S. Here is a sample to show the result, with no intention of publicizing the video.