The letter behind the name does not matter, both sides support it. There are two types of wars, real wars and proxy wars, real wars are host country and enemy country fighting. Proxy wars are funding and arming rebels in x country while enemy country does the same, using others to fight our wars for us.
Reagan daddy bush clinton and obama used proxy wars, while other wartime presidents use real wars. Iraq and afganistan for bush, vietnam over multiple administrations, though it hit it's peak under lbj and nixon, though this was also a semi proxy war against russia, as too was korea.
Proxy wars like what obama did are wars done when the american public is tired of supporting endless war, only difference between bush and obama was bush sent our troops to fight while obama threw money at another country to get them to kill eachother, equal human suffering, just different forces doing the actual fight.