JS: "If you pinch me under local anesthetic and I tell you it does not hurt, would you say that in fact there was a pain there, but that I just could not perceive it?"

Of course not. I would say that (1) you felt no pain. (I've been struggling with others who seem inclined to countenance "unfelt feeling" in this Forum!)

But (2) perhaps there was some unfelt nociceptive function going on anyway, not affected by the pain suppression, so some part of your somatosensory system may still have had some response to the pinch. (The big question is why all biobehavioral function is not unfelt function, like that.) 

And in addition, of course, since you were awake and saw the pinching, (3) you did feel something when you were pinched: it just wasn't somatosensory and it wasn't pain.