Perhaps I would risk using two words to what I think might simplify the 680 level settings: overload and noise. Please do not forget this is an oversimplification I'm doing.
Overload usually happens at the first stage, due to wrong gain settings. In the 680 the only gain settings are High and Low. Dynamic mics usually need High gain. Care should be taken with condenser mics, as some need High gain and others Low gain.
Noise usually happens when levels set by volume pot are mishandled. Shure had to go deep into this with their portable mixers, and explain quite well how to set levels, because users opened their level pots too much, and made a fame of Shure mixers being noisy. Later on they improved on this, by using a pot that set gain and levels together. But people still misused the Master pot. Peak leds helped warn you of overloads, and the 680 also has clip warnings.
I don't think distortion levels are increased if the gain levels are set too low. What does increase is noise levels, which might be considered some kind of distortion too, in extreme cases.
One good thing about the Tascam is that the level setting (Trim) is after just one chip stage, so at least it doesn't add up the noise of another stage. The other good thing is that there is a way, explained in the manual, that shows when distortion is happening at the input or further on in the chain. So it's a question of setting the gain level and see what happens below on the meter, and then setting the volume pot and see that the meter levels don't surpass the upper line.
But..., and on this there might be people who disagree, I believe eventual overload peaks for the first stage won't always be something bad and should help keep noise low.