Healer here, can't really help except to say read Kabolt link. I have healed many HM FP, HL LI and Operations with the person that wrote that write up. While the Shadow Tank is not his main, he is the reason that as a healer I personally believe the shadow tank is the best tank in the game (or at least my favorite to play with).
No real way to know what the problem is without being there, but this statement at least tells me at least part of the problem is you.
I assume Resilience is not really useful
Even on my dps shadow I know Resilience is Very useful.
Personally if I were tanking, I would also read write ups on all the different healers so that you have some clue of their abilities. As a healer i have done that with the other healing classes, the tanks and dps, so that I am better at my role as healer or dps.
I've told the healers (because I ran it multiple times) to focus healing on me, and I presumed they did. I mean, if the DPS need healing on that boss they must be colorblind or something, anyone should be able to spot those yellow marks on dee floor..
Any decent healer is going to focus on the tank, but none that I know are only going to heal the tank. I heal whoever needs a heal whenever they need a heal. I would agree with you and then do it the way I was taught no matter what the tank told me. I know a shadow tank can pretty much become immune to damage for a few seconds, so if it is between hitting the tank or hitting the highest output dps, I will hit the highest output dps, because at least with the shadow that wrote the article Kabolt quoted, he will easily survive until after I hit dps with a big heal.
Then how come I was able to survive the first 3 times I came around there?
First three groups may have all been good, over geared players that knew the instance and their class. My healer is well enough geared now that I could heal anyone through that fight pretty much no matter what they do. That does not mean a healer that can't is bad, it just means they are limited by gear. Same could be said for any role. It is just other than people getting knocked off the platform during Sav-Rak I consider any death during HM LI my fault no matter who the other players are.
Unless you know their class and their limitations, I would be careful blaming others, I would be more worried about making myself the best that I could be until I was able to carry people through. I have learned more from bad groups than I have good groups. You get in there with 4 really good players and you breeze through, but when do it with not great players, it teaches you to get better to get through the instances. Dying isn't bad in this game, if you learn something that will help you get through it more efficiently next time.