Normally I don’t give advice, but you asked. I have been burned enough by “friends” that I don’t put up with the kind of treatment she’s giving you. You will have to decide if you want to continue with the put downs and disrespect she’s giving you in return for...what?
Absolutely I would not go in with her in teaching a class. Ever.

Edited to add: One true friend who builds you up instead of tearing you down is worth ten who minimize and denigrate you. I know from having been there.

have you two ever taught a class together? if yes, how did it go?

Actually we have taught several collage classes together. We have different methods which we both share in class. Our students seem to enjoy the options. The only sticky part is when we discuss the Christmas village quilt class and she refers to my method as collage and not applique.

I wouldn’t take it personally, as I don’t feel it was meant that way. Sounds like she is a purist who believes appliqué should involve some sort of sewing. Otherwise it’s just fusing. Maybe you should be teaching separately since you have such differing viewpoints on appliqué.

I'm trying to wonder why she cares so much. Sounds kind of silly to me.

just stand up when it's your turn and say this is not considered collage and go from there. Good luck!!

you do raw edge applique. she does a turned edge applique (whether it is hand done or machine done). I would continue to call your work applique and be proud of the work you do and teach!

Nancy Zieman has shows of landscape quilting using the raw edge applique and the pieces quilted down in the way you described.

You answered the question! Both suggestions are good, although I lean towards the first. This solves how to promote the class without losing a friend who happens to see only in rigid black and white in this area. You found a way to not put a wedge in the friendship by being respectful of the friend’s shortcomings/weakness.