That's not the full passage, though. The full passage characterizes what must be met to declare intent: Check that a card can be played. Pay for it.

This makes no mention of choosing targets. As long as there could be a target during this timing, you don't need to pick now.

I'm not trying to be pedantic here. Declaring intent to play something doesn't mean picking targets. Indeed, that's not one of the steps.. Most games that have timing for when things are targeted call that out.

The reason this game doesn't call it out during initiating the action is that a card's text executes in the order it is written.

As far as I can tell, the order looks like this... Declare intent to play: Prove it can be played. Pay for it. Resolve consequences of costs. Suffer an AoO if needed. Resolve the effect of the card, which is the body text in order.

I don't see textual support for intent including picking targets, merely that you must prove resolving the card would change the game to the best of your knowledge.