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Hamilton was the commanding officer of a the NY Artillery Regiment at the time of Trenton; further, he had been ill and left a sickbed when Washington began to cross the Delaware, so intimate field action as shown would have been unlikely. He was instrumental in that battle setting up his cannon at the top of a hill in Trenton and raining cannonballs down at the Hessians.

Yorktown was different. By that time Hamilton had been an aide-de-camp to Washington and had had a falling out with the General since he wanted to get more battlefield action. At Yorktown, Washington gave Hamilton the honor of commanding three infantry battalions (he had, by now, left artillery and was a light infantry officer) to march against the toughest British position (redount #10).

It is interesting to note that Hamilton's original NY Artillery survives now as the 1st Battalion, 5th Field Artillery, and is the oldest unit in the US Army. As you likely know, after the RevWar, the army was disbanded. The only exception was the NY Artillery regiment since they were assigned duty at West Point to guard the arms deposited there.