The DM is trying to do something reasonable, and doesn't know how to do it.

I try to avoid using out-of-character terms when I'm speaking in-character. My Ranger Gustave doesn't refer to his "Guisarme +4", but he does speak of his magic guisarme. But I refer to it when necessary, during combat. "OK, that 5 points of damage, +7 for STR bonus, +4 for the guisarme, +5 doubled to 10 for the power attack..."

Similarly, the Fighter 2 / Ranger 4 / Horizon Walker 6 calls himself a "ranger".

So I sympathize with the DM's goal. But what he's doing doesn't succeed in reaching that goal. Even with care, we cannot avoid referring to the game mechanics, if we want to tell the DM that his monster took 23 points of damage.

So the important skill for successful immersion isn't self-consciously trying to avoid breaking character, it's the ability to get back into character quickly and easily.

I find attempts to get rid of all D&D terms to be far more focused on the mechanics and not on the fantasy world. I'd prefer to say what needs to be said quickly and get back to the battle with giants. The attempt to rephrase a gaming mechanic into the fantasy milieu, so the others have to try to translate it back to the game mechanic, is far more modern and intrusive.