SD card and Studio One/DAW Mode are completely different things. You CAN play SD tracks back to the mixer (the command in the menue changes all inputs of the mixer automatically) and mix the tracks and possibly record Main L/R on SD card, but this happens outside (!) Studio One.
If you want to use Studio One, you have to transfer the recorded files from SD card to your computer, which needs a SD card slot in your computer or some kind of adaptor. It is also possible to transfer the "scene file" (the mixer's settings) from SD card to Studio One, but it makes more sense to start from scratch and import the WAV files (named 0, 1, 2, 3...) into Studio One, add track names, add processing, add effects, write automation etc.
DAW Mode simply means that what is happening in Studio One is "physically represented" on the StudioLive mixer. You're not using the mixer's processing (this is only an option). The faders and encoders "belong" to Studio One and simply do what the DAW does. Moving a fader by the mouse is replaced by moving a fader on the mixer, but all audio is handled in the computer, not in the mixer.
Hope this helps to see things a little bit clearer... ;-)