4H would be the correct designation in this case. The height, according to the web site, is the highest point between the front two axles. A 13' high travel trailer would thus not be a "high" vehicle unless the tow vehicle were tall.
I think the cutoff is actually 7'6", but I may be remembering incorrectly. Their EZPass Motorhome plan (which doesn't apply to 5th wheels, unfortunately) is a nice deal in that they'll charge the short rate rather than the tall rate on the thruway, which for me at two axles works out to about half the toll.
On a per-mile basis, the NY Thruway is actually competitively priced with most of the other toll roads in the northeast. Obviously it's sometimes hard to make good overall comparisons when the vehicle class definitions vary as much as they do. (Maine charges more for dualies but not for tall vehicles; New York charges more for tall vehicles but not dualies.) Even so, it still stings a lot to go from one end of the state to the other and have to fork over the contents of one's wallet.