An expensive lesson was learnt the hard way today, because of my ignorance.
The Kindle Fire HD 2013 (Angular thing, speakers either side of the back.) is NOT a Kindle Fire HD 7, despite what its owner says, and does NOT appreciate having a KFH7 bootloader flashed to it.
By which I mean it now won't boot, and doesn't appear in the list of attached USB devices, even with the factory cable. So I've got to go unearth a JTAG adapter, and pray I can do something to recover it that way. (I'm sure I have the original bootloader somewhere; but it's whether or not I can feed it to the device in a way that'll work, that's giving me kittens.)
I only learnt of the difference between the two when I went to go find a teardown to see if it said where the EEPROM was soldered on.
BTW, when the tutorial for rooting says it doesn't work on a particular model, it might be a very, very good idea to go double and triple check you're not trying to root that model it doesn't work on.
IMPORTANT: Apparently, the hardcoded 1st stage boot ROM -- the one that loads the bootloader from flash/USB -- checks the file hash of the 2nd stage it's going to load; and if it doesn't match with what it expects, it will not play ball. This is a Bad Thing™.