Yep, I agree. There's a reason everyone was saying "the war in Afghanistan and Iraq", "Vietnam war", whatever you want to call those "conflicts", they were wars.

It is. The President is the Commander in Chief, who outranks everyone in the military, orders them to do things, all that. Congress controls spending and has the exclusive right to declare "War", which the President then uses to act with more power than under normal circumstances.

Sure, which they're allowed to do that. These definitions of "War" can be wrong.

Exactly. There are limitations to what the President can do as the Commander in Chief, but sending troops to places within the restrictions of the defense budget or what Congress authorizes through what has lately been the AUMF (Authorization of Military Force). These things are not "War", however.

If you want to get technical, the US has not been at "War" since WWII. They have been in numerous "Military conflicts" or "peacekeeping operations" or whatever you want to call them, but it's not "War".