When you're "driving the signal hard", you're risking pushing the amplifier into overload. At which point, most amps will clip the signal. Depending on what kind of amplifier it is, it will either clip hard (transistor), in which case it won't sound warm at all, or will clip somewhat more gently (MOSFET, valve), in which case it won't sound so nasty. But the sound of a domestic amplifier being overloaded, whatever the owners of Fender guitar amps might think about theirs, is rarely ever going to be "warm".
As for the analogue vs digital thing - I am pretty well convinced that it's a subjective perception thing, more to do with people's appreciation of the rituals and history of listening to music on "analogue" media than anything else. It's interesting, incidentally, that you never hear anyone waxing (ahahaha) lyrical about the "analogue" tape quality over digital - it seems to be confined to LPs.
Whatever it is that's nice about listening to analogue - LPs or otherwise - I suspect it's got very little to do with the quality of sound reproduction.