Hi,
first of all,
Quote: What I am trying to say here is both circuits are different and hence all the parameters are diff(ex gain=2X), that's clear.
I would say that this is not clear. The gain is still the same, only the excitation has changed. Anyway that's unimportant and you can continue using whichever definition you are most comfortable with.
See my attachment. IMO, the problem is the way you are splitting up the circuit. The differential amp naturally splits down the center to go from differential to single-ended in the voltage sense. For current, you would have to split it differently.
regards,
Aaron