You’re agreeing with me that they’re using HAT incorrectly, yes?
It would help to know how your check airman would teach it to you. Height and altitude are not interchangeable words, despite their similarities. It's the same thing as weight and mass. They're two very different, but similar, words, often used improperly. You've pretty much pointed out your correct understanding of the intent of HAT in your original post.
The reason I ask how your check airman would teach is that you state the definition in the manual is "height of aircraft above TDZE" which, although not 100% correct to the regulatory definition, would be correct if the intent of their phrase is to imply the DH (they should just change the wording in the manual if this is the case). One of the hardest types of discussions I have with other pilots is where we are both arguing the same point, but neither of us realizes it. Could it be that your check airman is arguing that the company means the DH, but because in discussion both altitude and height are used, that everyone is getting confused? I'm not saying the check airman is right. I'm a check pilot in Canada, and it's not a delegation that immediately confers instant knowledge of all things aviation, despite the facade we sometimes project. What I'm saying is that if you're both arguing the same point, it could be getting lost in syntax.
If it is really that bothersome and you think there is a safety concern, I'd file an SMS report if your company is one of the Part 135 operators that require an SMS. If not, is there a chief pilot or another check pilot you can talk to?