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@Airhead
I think it's more the difference between optimum and maximum input. The problem being distortion as the amp approaches the edge of its performance envelope. Pretty much how you've summarised it. I might have said 'can' rather than 'will' overload.
Obviously I trying to simplify it grossly and don't want to get started on preamp colour and distortion either although preamp distortion is hardly an unknown in the recording process.
The think the process you are describing is called interpolation clipping, and yes some dacs do have headroom above to cope with it but technically there should be no headroom above 0dBFS since it is supposed to represent the maximum digital level.