They aren't the same thing at all. We've had Presidents ask for authorization without asking for a declaration since the 18th century - the Quasi War was the first I believe.

The precedent is well understood. The President may ask for authorization for any extended war and for a formal declaration, if desired. Then, and only then, will Congress act. Congress will not issue a declaration absent being asked by Commander of the military, for obvious reasons.

This idea that Congress is somehow not doing its job because it's not issuing a formal declaration that were not requested nor required, is simply nonsense.

Frankly, if requesting authorization was the same thing as requesting a declaration, then one could just as easily argue Congressional approval of funding for a war is a declaration.

A declaration of war or an authorization of force are both Congressional approvals for use of military force against an enemy of the state. I get that's inconvenient for your argument, but they're the same.

And the president doesn't have the authority to declare war on his or her own accord, full stop, because the constitution explicitly gives that right to Congress (and no other branch).

Any convoluted timelines around requests are immaterial to those facts.

If the president uses the military to attack an enemy of the state, without Congressional approval, that's outside of his or her authority.

Not for purposes of the Alien Enemies Act they aren't the same.