The only thing that gets sent from the piano to the computer is key velocity. So the "harness" settig on the piano is sent only indirectly beacuse it determines how your key presses get mapped to MIDI velocity numbers.
Why is it brighter sounding when played from the computer screen with a mouse? I think it must be the velocity. But could it also be the you have moved? Are you setting at the same location relative to the speakers? Listening geometry matters more than most people wold think.
There is a lot you can do to adjust the sound. The Alicia Keys samples comes with and plays inside Kontakt Player. This is the free version of Kontakt 4 that they bundle in there for you. It is stripped down but still has lots of features and you can adjust brightness, reverb and so on.
Also you can use Kontakt Player as a plug-in inside a DAW. That's how I use it. Then you have a full range of effects you can use to color the sound, graphic EQ, parameteric compressors, better reverbs or what have you.
If all you are doing is audio playback, not recording from a mic or line-in then you can get an inexpensive USB audio interface and, yes it might soud better, Some of these interfaces have MIDI too. Technically this has no effect but it can help reduce cable clutter. And importantly save th little 3.5mm audio jack on the computer from damage
One like this might work
Alpha One other advantage of the box is you get real knobs to control the volume
If you are going to record or you need MIDI the price at least doubles
