When, diplomatically speaking are we at war with a country?

I’ve been very busy the last few weeks and have left posts that I made unaddressed, and apologies for that, but I am following the news I simply don’t understand how international relations work.

There are some very good arguments, pro and con, against our involvement, in whatever capacity, in the terrible invasion of Ukraine. But I don’t understand the distinction that has been made.

We provide anti-tank missiles that kill Russian soldiers, but Biden refuses to even allow Poland to provide airplanes. The idea that we might lob a few cruise missiles against advancing Russian troops is considered a clear act of war. I don’t understand this sort of differentiation.

Is it bad if my neighbor shoots at my house, but better if his brother gives him a gun with the purpose of using it to shoot at my house? Is it even more different if say (like the comparison between anti-tank missiles and airplanes) if my neighbor’s brother only gives him a bolt action rifle and not a semi-automatic one, again, with the express authorization to use to shoot at my house?

It seems to me, again, without debate on what we should or should not be doing, that these things are so similar that the distinction makes no sense. Am I off base?

As a follow-up, I think this is pretty similar to the Vietnam War. Russia and China were supplying the North Vietnamese with arms to kill our guys. Is the realpolitik answer that we engage in that fiction so that we can fight proxy wars without anyone needing to thump their chests and risk nuclear war?

Even if that is the case, what explains the airplane/anti-tank missile distinction?

I stated this incorrectly for the analogy. If my neighbor’s brother shoots at my house, should I be less mad at that man (neighbor’s brother) if he simply hands his brother, my neighbor, the same gun so that the neighbor can shoot at my house? Is the brother less culpable because he didn’t shoot at me directly? Criminal and civil law says absolutely not and any judge would laugh if you made that argument. Why does international diplomacy consider it to be different?