<<That section defines what is needed to introduce combatants into military hostilities (ie war). Have you read it?>>
Of course I read it. The point is that there can be military hostilities which are not, technically, war. As in Korea - - uniformed armies were shooting at each other. You would say, "hostilities (i.e. war.)" But of course you'd be wrong. They were hostilities, there was no war. It was officially a "police action."
<<Why does the "War Powers Resolution" not refer to the power for declaring and maintaining war?>>
Probably because the power to declare war is clearly set out in the Constitution. The War Powers Resolution would be about "War Powers" not defined in the Constitution, i.e. the power (falling short of a declaration of war) to send U.S. troops into combat-like activities. Giving the President something that would permit him to start wars without having to go through the technicalities demanded by the Constitution for declaring war.