Because your actual armed forces aren't directly involved. No NATO boots on the ground, no NATO pilots flying sorties, no US cruisers conducting shore bombardments. OUR forces aren't the ones firing the actual shots.

That's it. That's the only difference.

That's why it's called a "proxy war." Instead of sending your own troops, you're sending materiel, money, training, and intelligence to support an allied country against a rival major power. ETA: And yes, we are accomplices. Accomplice is just a pejorative synonym for ally.

Because major powers don't want to fight each other directly due to the risk of escalating to World War and the probable nuclear exchange WW3 entails. And aiming blistering rhetoric (and diplomatic weapons like sanctions) at a rival for providing material support to the opposing belligerent is NOT the same as declaring war on them.

It's how basically all the wars of the last half of the 20th and now the 21st have been/are being fought. Because the alternative is worse.

ETA2: But yes, countries can and have declared war on a 3rd party for providing material support to their original opponent. You just have to be very careful about doing that when both you and that 3rd party in question are nuclear-armed.