I assume you mean instrumentation amp rather than opamp but yes, you can just as easily correct offset at the input as the output.
At the end of the day the output is all that matters. The offset correction circuit can do anything that nulls your offset at the output.
I can't read any sense in the idea to vary VCM.
The OP is talking about an inst amp and I assumed because of what they said they meant the "ref" pin when referring to "VCM from output". Possibly a mistake?
"However, the input offset voltage defines the minimum input voltage that can be treated amplifier"
I don't agree. It's common to look at signals that are smaller than the offset and it shouldn't matter where you cancel that offset.
In gain of 1000 example you might have 1mV of offset giving you a 1V signal with 0 input. Now cancel it at the output -> 0V signal. Now an input signal of 0.1mV comes in and the amplifier output will increase by 0.1mV*1000=0.1V to 0.1V out. That's perfect.
If your gain is variable and input offset dominates then there might be an advantage to canceling at the input. Now if you change gain the offset will remain mostly cancelled. But otherwise I'd still suggest its an implementation choice: where is it easiest to inject your own offset.
The term VCM is just confusing related to the circuits. They don't have (or at least don't show) VCM control feature. And no CM reference voltage to set the output VCM. In so far it's unclear what you mean with "adjusting common mode output voltage".
The offset can be manipulated by injecting a differential voltage or current between Vref+ and Vref- in your circuits, not by shifting both in the same direction.
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