It wasn't the war though. The war began with the non-aggression pact between Bajor and the Dominion and the attack on DS9.
That would be like saying that the incursion of the Defiant into Dominion space in "The Search" was also the war.
Since "The Search" happened immediately after "The Jem'Hadar", then yes, Starfleet would be on a war-footing at this point. It had gone hot and you can't put the shit back in the horse once first blood is spilled. Sure, there was a short cool-down period when the two powers were sizing each other up and moving their pieces on the board, but the die was fully cast at that juncture.
As for when the war officially started, was there ever a formal declaration of war between the two powers? Such a
political maneuver usually happens
after a kinetic military engagement where lives are lost. From Memory Alpha:
Following initial Alpha Quadrant expeditions into the Gamma Quadrant, rumors began to filter through of a ruthless race of conquerors known only as "The Dominion".
In 2370, first contact was officially made between the Federation and the Dominion, when the Jem'Hadar destroyed the Starfleet vessel USS Odyssey. A state of cold war rapidly developed thereafter between several of the Alpha Quadrant's major powers and the Dominion. A Dominion invasion was widely regarded as inevitable, as the
Founders were driven by a
xenophobic need to exert absolute authority over all other civilizations.
I personally draw the line in the sand at the first kinetic engagement and loss of life when it comes to the exact point when a war starts. The United States entered into a state of war with Japan upon the attack at Pearl Harbor on Dec 7 1941. However, by the logic you stated, the official "war" didn't begin until its declaration by Congress a day later on Dec 8, even though we were forced into a hot war footing only 24 hours before. Did the lives of those lost at Pearl somehow mean less than those who died
after war was declared? And since no other war since WWII was officially declared by Congress, does this mean that Korea, Vietnam, Gulf Wars I & II, Afghanistan, etc., didn't exist either? Not important enough? The hundreds of thousands who died in all those conflicts are irrelevant because those weren't "real wars" by conventional definition?
Sorry, but this gets into a realm of pedantry that is desperately (and weakly) grasping at defending the original premise to support the (objectively incorrect) fact that no Galaxy-class ships were destroyed on-screen during the war. Such a position simplistically and blatantly ignores the dynamics involved in the conditions of international armed conflict.