Well, you never get the feat, but you do get an identical benefit (the removal of the penalty).
That said, the text is "you can shoot or throw a precise weapon." The weapon you are using has to be the one with the property or you get no benefit. I really don't see how you think it's "obscure" at all as it's really quite precise (pun not intended).
Full text:
You can shoot or throw a precise weapon
at an opponent engaged in melee without
incurring the standard –4 penalty. This
benefit does not apply if you already have
the Precise Shot feat.
1. It's very clear you don't get the feat. You don't even get "the benefit of the Precise Shot feat." You get an identical benefit (removal of the penalty for firing into melee). It doesn't apply if you have the Precise Shot feat because that feat already removes the penalty, so this property no longer has anything to do.
2. It's very clear it has to be the weapon with the Precise property making the attack to get the benefit.
