score:7
Does that mean I must go through a third country when traveling from Israel to the US ...?
No.
If the answer is no, then what passport information should I fill in the airline check-in form - The one I'm exiting the country with, or the one that I intend to use at the destination?
The one you intend to use at the destination. The airline doesn't care how you leave the country, they only care that you can enter your destination (otherwise they'll have to bring you back on their own dime).
In Israel, you'll use both passports. You'll use your Portuguese passport to check in (they'll also want to see your Israeli passport, as that's where the sticker will be), and you'll use your Israeli passport to go through the security and the passport control (that doesn't have any stamps anyway).