Fortunately, if someone is in your house, most courts will let you assume they are there to harm you, not just to steal from you.
Most yes, the ones in NY, No.
I think warnings are legally optional in New York, my state - it's not mentioned in law that I know of.
My state too and you are correct warnings are optional in NY but the legal situations for shooting are VERY narrow, essentially leaving little choice than to give a warning unless the threat is EXTREMELY severe. NY is essentially a "reasonable belief of imminent death/substantial bodily harm state and it is clear that a simple breaking and entering will not qualify in the eyes of the court.
For what it's worth, here's my take on the situation (BG with an arm load of stuff). I'm going to tell him in no uncertain terms to stop (exact verbiage optional), with my gun pointed at him (finger off the trigger). If he stops, freezes, drops the stuff, runs out the door, jumps out the window then we go to the next step of either his surrender and/or 911. If he makes an aggressive move, runs toward me, drops things and reaches in a pocket or points something at me- I shoot. At that point the BG has escalated the confrontation to a physical one, not me.
As for the argument of "Now he knows you've got guns/valuables/whatever." Well, I guess he does and he MIGHT come back, which would suck, but to shoot someone over what MIGHT happen is over the horizon on my moral compass. I'm not saying you should or shouldn't. I'm saying I won't.
Still happily answering to the call-sign Peetza.
You do not HAVE a soul. You ARE a soul. You HAVE a body.
He is no fool who gives what he can not keep to gain what he can not lose.
-Jim Eliott, paraphrasing Philip Henry.