After playing around with the SWAT class for a while and then playing with a without perks, apparently the wording of the perk is misleading.
you only take Health damage from sonic attacks
does not mean sonic attacks don't deal armor damage anymore, as they have never done armor damage. What this really means is that the only source of Health damage is from sonic attacks while you have armor on.
Thus, what Heavy Armor Training does is give you 50% armor to begin a game with. With armor on, you will be unable to be grabbed by Clots and you will not begin to take Health damage until you no longer have armor on, with the exceptions being Sirens' sonic attacks (and fall damage).
Others have been able to confirm this behavior. See this thread:
HAT prevents damage to health from non-sonic attacks but description is confusing until you actually use perk and notice effects.
There are some other things that dip straight into health damage. From the top of my head, I can only think of fall damage and Hans drains.