Little late to the party here but you're wrong. Unless something is being done to the signal (eq, other processing, etc) then you want the digital side to remain at max volume (i.e., no reduction in bit depth). The audio will have been mastered close to 0 db with little or no clipping (exceeding the maximum amplitude possible) and should not be "too loud" for your amplifier. If it is there is something wrong with the gain configuration of your setup.
As discussed, lowering the volume in software (which is what you likely mean by "controlling the DAC") is accomplished by reducing bit depth. By definition you're reducing the amount of signal present which makes your S/N worse. If by "controlling the DAC" you mean reducing the signal at the DAC's analog stage then there's nothing theoretically wrong with that (except that it's rarely if ever the case in computer sound cards or hi-fi DACs) as it is the equivalent to adjusting the volume on your (integrated) amp.
Probably the most common exception to the above is people using a digital EQ to attempt to improve the sound of a mediocre computer audio setup. Most digital EQs allow for both "boost" and "cut" -- increasing or decreasing the amplitude of individual frequency bands. If you EQ the music the proper way you should only cut frequency bands in order to emphasize the other bands to suit your listening tastes. For example if you want to try to increase the amount of bass you should cut the mids and the highs and turn up the volume (on your amp) to compensate. EQing this way ensures that you're not forcing the signal into clipping by boosting too much.
You still won't get great results because if you're lacking bass or highs then it's most likely because you're not running full range speakers. If the sound does change when you EQ it's most likely due to increased distortion, not because you're appreciably increasing the amount of bass present. That's not true for full range speakers but if you have full range speakers you probably wouldn't need to EQ in the first place. (Full range speaker are usually 20 - 50hz to 20khz.)
If it's still not clear than think of it this way: Standalone CD players don't have volume controls. Their output will be "max volume" of what's on the CD. You can think of your computer and sound card like a (crappy) CD player. Make sure it's at max volume and adjust the level on your external amp or amplified speakers.
'As discussed, lowering the volume in software ... is accomplished by reducing bit depth.'
No, reducing the amplitude of a digital signal does not change its bit depth. If the amplitude of the digital signal is reduced, then a couple of bits on the MSB side will be constant. However, reducing the bit depth would make bits on the LSB side constant.
To state it differently, a lower bit depth increases the quantization levels and results in lower quality audio. Reducing the amplitude retains the same quantization level, and therefore has no reduction in audio quality (At least, not due to bit depth, it does change the SNR as you suggested)
You're correct that lowering amplitude ~= reducing bit depth. I'm not sure that lowering amplitude digitally keeps S/N constant but I'm a bit out of my depth (excuse the pun) at this point.
To further muddy the waters, in most cases using software volume control won't noticeably reduce audio quality. The main point I was trying to make is that on most computer setups 100% volume should be the least noisy, most accurate signal that it's capable of.
As discussed, lowering the volume in software (which is what you likely mean by "controlling the DAC") is accomplished by reducing bit depth. By definition you're reducing the amount of signal present which makes your S/N worse. If by "controlling the DAC" you mean reducing the signal at the DAC's analog stage then there's nothing theoretically wrong with that (except that it's rarely if ever the case in computer sound cards or hi-fi DACs) as it is the equivalent to adjusting the volume on your (integrated) amp.
Probably the most common exception to the above is people using a digital EQ to attempt to improve the sound of a mediocre computer audio setup. Most digital EQs allow for both "boost" and "cut" -- increasing or decreasing the amplitude of individual frequency bands. If you EQ the music the proper way you should only cut frequency bands in order to emphasize the other bands to suit your listening tastes. For example if you want to try to increase the amount of bass you should cut the mids and the highs and turn up the volume (on your amp) to compensate. EQing this way ensures that you're not forcing the signal into clipping by boosting too much.
You still won't get great results because if you're lacking bass or highs then it's most likely because you're not running full range speakers. If the sound does change when you EQ it's most likely due to increased distortion, not because you're appreciably increasing the amount of bass present. That's not true for full range speakers but if you have full range speakers you probably wouldn't need to EQ in the first place. (Full range speaker are usually 20 - 50hz to 20khz.)
If it's still not clear than think of it this way: Standalone CD players don't have volume controls. Their output will be "max volume" of what's on the CD. You can think of your computer and sound card like a (crappy) CD player. Make sure it's at max volume and adjust the level on your external amp or amplified speakers.