I personally would be rather flustered and perhaps unintentionally brisk, if a person I did not know started lunging a horse I did not know in the arena while I rode. My horse is greenish and very sensitive. Lots of clucking and line flipping paired with a horse that doesn’t lunge well could result in me wrecking out. Therefore I might behave in a manner more oriented towards me staying alive and less oriented to your feelings.
My last boarding barn had a no lunging in the arena if people were riding. The barn before did not. And they should have.
Have you clarified the rules regarding lunging in the arena? Particularly if others are riding?
When I was first backing my horse, I did a fair amount of scheduling to get in the arena with as few folks as possible. You might consider similar if you are trying to teach your horse to lunge. Sounds like your horse may have a more “western” concept of lunging with the turning in to face you.
I encourage you to try and project more positivity. Catch more flies with honey than vinegar and all that.