CB is correct. "ce" refers to the actual document, or right. "se" is the act of granting the right.
As for replacing the US cardboard...It looks like they want me to go throught the whole process of verifying my foreign licence again! Surely can't be true, since I only did that last year, and they know now it's genuine! It's not the hassle that's the problem here, but the requirement to present myself at a FSDO! I think I'll stick with my bit of cardboard
