further to Oli and Digby,a division was a tactical unit that was capabale of operating under the officer or NCO placed on the flank (see the quote that you raised). When the squadron was formed it was equalized into two companies, with each company also subdivided. As with the infantry period equalization was required to redistribute for uneven casualties.
Three's were used to enable a division of any size in line to perform operations not line and to face about. For example, a division could form column to a flank, take ground to that flank, then advance to face the enemy again. This was an important manouver in order to extend a units frontage and threaten the enemy flank. If you consider that wheeling a squadron with 60 files would be a nuisance, then consider Ney's "Quick Manouver" you can see that he adapted cavalry unit tactics (which was his background) to the infantry.
For facing about, that is not the simplest task for a single horse and rider, and it is impossible for two riders boot-to-boot to be able to turn on the spot. The cavalry solution was to advance with each Three wheeling about its right hand rider. It was very quick to execute and when repeated faced the unit in its original direction again. Should a squadron or lesser division suffer a reverse it was necessary to go "Threes About" with utmost rapidity and ride to a distance at which the commander judged he had created enough time and distance to go about again and front the enemy. If the regiment was in column, in theory, the squadron suffering the reverse would pass to the left and right of the next squadron (hence one of the reasons for ensuring a significant gap between squadrons in column. This enabled the next squadron to charge the pursuing enemy, breaking them in turn.
So a division responded to the word of command of the officer or NCO placed in command of the division and functioned as a tactical unit, whereas threes were a functional group driven by the shape of a horse and the dynamics of how they moved.