Obama is not "declaring war" despite your effort at semantical gymnastics. No more than GWB did.
Let's face it - it has been a long time since wars (military engagements - whatever words you wish to use) were begun with congressional declarations of war. I think WWII was the last one. Congress itself gave the president such a unilateral right with the War Powers Act and the various resolutions granting GWB the power to fight terrorism.