Next time I go into the post office I'll take a picture of their question on the pad, I don't believe it mentions weapons.
The language you highlighted in red says they MAY check to see if it is unloaded. That seems to be the crux of the argument for those who say you must declare firearms. But there is ambiguity there; if you don't tell them what it is, how would they know to ask? I gather by policy or law they're not allowed to ask specifically what is in a parcel. I believe the boiler plate question is whether the parcel contains anything "liquid, fragile, perishable, or potentially hazardous."
Reasons you wouldn't mention that you're shipping a firearm would be to avoid an untrained clerk freaking out and demanding your driver's license, firearm permit, the firing pin be removed, requiring you to fill out a 1508 form as if you were an FFL, making you measure barrel length, stealing your gun, etc.