If you choose to interpret the War Powers Clause in the Constitution as referring to war, and thus to your list of definitions, that may have been what the Founding Fathers wanted. But in the current interpretation of the text, which does not actually mention war per se, the War Powers Clause does indeed refer to military action.
What I'm trying to say is that despite this, another part of the same document, albeit on a different page, says that you can ignore that bit if you sign on it with another country. In case you're wondering, that's how the Controlled Substance Act works. If treaties couldn't supercede the Constitution, then the federal government would have no way to prevent anyone from smoking marijuana drug, opium drug, or ingesting cocaine drug. Many other matters of intra-state trade control (which the federal gov. is not allowed to do, Constitutionally) are resolved by using treaties.