I say no difference. If the output side of the amp is being pushed into clipping, the amount of input voltage used to push the amp to that point is irrelevant.
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Your logic has led you to the right conclusion. The amp is on full power all the time and is only responding to the strength of the signal you are feeding it. Reducing gain of the source or turning the volume control effect the same end result - less signal to the amp and a lower db level. Sounds like you need a more powerful amp.