That's because the United States official position in the civil war was that the south was a part of the US that was in rebellion, and not a sovereign state that we were at war with.

I don't see how that negates the action of sending of troops for combat. You're just arguing an excuse as to why congress wasn't sought to to declare war in this "case where the President decid[ed] to send troops." There is always some excuse for that nowadays, quite conveniently, so you're in good company.