Richard III

Is Richard still as determined in his cause, or when he says "a horse, as horse, my kingdom for a horse" has he turned his back in a cowardly final act?

I see nothing in Richard’s words or actions on Bosworth Field that hints at cowardice. Catesby offers to help him retreat from the battlefield, but Richard declines, retorting:

Slave, I have set my life upon a cast,

And I will stand the hazard of the die.

He asks for a fresh horse in order to maintain the attack.

With his various schemes and intrigues, Richard has rolled the dice. He now accepts his fate, either to prevail or to die in battle with Richmond. He seeks out Richmond, thrusts and parries with him, and is slain. That doesn't strike me as a coward's death.

Richard's speech here is exaggeration, like when we say we'd give an arm and a leg for something. He hasn't fought so hard just to turn and run.