That's incorrect. Sensory information from the cornea is transmitted to to the brain via the trigeminal nerve.
(Edit: You updated your remark while I was still typing.)
I think when they talk about sensitivity, though, they're talking about the ability to discriminate what part of the organ is being touched, or 2-point perception (i.e., if you move two pins closer and closer together, how far apart are they when you can no longer detect them as two separate points). The fingers are best at that.