Oops, thanks. I've always had struggle remembering these terms because they are so confusing. "Specific" means that something only relates to one particular element X in a set of more than one thing. Given that meaning, one would expect specificity to mean "how good is the test at correctly reporting the thing it is designed to report X rather than confusing a different thing (an element from the complement of X) for X", but that is sensitivity. Instead, specificity means that with respect to the complement: "how good is the test at correctly reporting all the other elements in the set ¬X as negative rather than confusing an element from ¬X for X".
"Sensitive" has a clearer intuition: For example when you want to detect a weak signal, then if your antenna is more sensitive, you can better pick up the signal, thus you are better at telling whether the signal is really there (you have a better true positive rate).
I cannot think of a better name for TNR, though, other than simply sticking with TNR and TPR.