It's like saying Buck had lost the battle but not the war.

BEATEN = defeated.  BROKEN = loss of spirit, heart (with a horse, we say that it is BROKEN IN to mean that it has been trained and can be ridden - in the past this was usually by way of force and whip to dominate them, now it can (and in my opinion, should) be more humane).

Basically - Buck is tired, he's had enough of fighting but his spirit is still there and he'll fight again another time - he wants to survive although he is tired.