My experience is from a new (april 2011 manufacture date) desire.
The best way to be sure is to contact HTC through their website's support form and ask them what kind of screen your phone has.
Sorry to contradict one of the super fonts of all knowledge on the forum, but from my experience there is now only ONE certain way to know what screen your desire has in it and that is to take it apart and google the serial/part number on the back of the broken screen. #
DO NOT TRUST THE BOX TO IDENTIFY YOUR SCREEN CORRECTLY !
DO NOT INVEST IN A REPLACEMENT LCD UNTIL YOU ARE 110% SURE !!!
DO NOT PLUG IN YOUR REPLACEMENT SCREEN UNTIL YOU HAVE CHECKED YOUR OLD ONE VIA GOOGLE FOR THE SERIAL NO.
I have just bricked my phone by following contacting HTC (with my serial no's etc.... which they said they could not tell me what type I had, and to check the box) and then following HTC's instructions of "look on the box" as there was no mention of AMOLED on my box, but the screen inside my phone was a Samsung AMOLED.... noticed too late...
phone = dead
HTC offer the manufacturer warranty, call them directly. As its cracked from the inside they should cover it anyway.
and similarly, No.... my experience contradicts yours also.... if you contact HTC they will inform you that broken screens are not manufacturing defects, even if it is only the internal LCD display that died.... and will not offer you warranty repair.
So to Sum up.....
A. Do not trust what the box says about your screen.....
B. Do not go ahead with a swap until you have googled your existing screen and made "extra sure"
C. Do not expect HTC to cover you with warranty for broken screens (even if just the internal screen is broken)