Yes, it's officially still the Department of Defense.
If this were a news outline writing "Department of War" I would be concerned. But in the case of the Anthropic CEO's blog post, I can understand why they are picking their fights.
It's a silly shibboleth, but I automatically ignore anyone who calls it the Department of War or Gulf of America. Hasn't steered me wrong yet. They're telling me they're the kind of people who only care about defending fascism.
I think it's worth giving people a tiny bit of grace on this. I've surprised people by explaining that the "Department of War" is just fascist fanfic and that the legal name has not changed.
It's a testament to the broken information ecosystem we're in that many people genuinely don't know this. Most will correct themselves when told. I agree with you that those who don't are not worth engaging.
I would not defend all of Google's decisions in the Trump era, but complying immediately with politicized name changes has always been the status quo. Even in healthy democracies, the precise names of geographic features can be extremely controversial, and no sane company wants to get in a debate with the Japanese government about the real names of various islands.
They can, however, rename their Twitter/X accounts and vacate the @SecDef handle, which seems to be up for grabs now, if anyone wants to do the funniest thing...
No, fighting a war requires only engaging in international armed conflict.
Declaring a war requires Congress, and fighting a war other than in response to an invasion may be illegal under US law if Congress has not exercised its power to declare war, but that doesn't prevent wars from happening it just makes it illegal (though the only actual remedy is impeachment) for the President to wage war without authorization. And, in any case, that’s largely moot because Congress has exercised that power in an open ended (in terms of when and against whom) but limited (in authorized duration of any particular action without subsequent authorization) manner via the War Powers Act, giving every President since Nixon a blank check to start wars with full legal authority and then allow Congress an opportunity to vote to pull support from forces already in combat and hope the enemy already engaged is willing to treat the war as over as the only after-the-fact constraint.
Of all the silly things that Trump did, I think this one is the most reasonable. This has always been a department of war. Calling it defense was propaganda.
After it was changed from DoW the first time (in 1947), it was called the National Military Establishment (NME). They renamed it in 1949, potentially because "NME" said aloud sounds like "Enemy"