It also sounds like the friend had the opportunity to sign the liability waiver, but didn't. So the boarders asked you to give their daughters the lessons.
Personally, I'd tell the friend to take it up with the boarders. It is their decision who teaches their daughters and I don't think you should get in the middle of that decision.
But I definitely agree that if the boarders change their minds and want the friend to teach, do NOT allow it without that waiver signed. It seems to me you were pretty clear on the requirement and the friend was trying to circumvent your policies based on the fact that you rode with her a few times (as you said). And now that you are being paid to give the lessons, she is feeling put out.
The only thing I'd 'discuss' with her (the friend), and it would be a short discussion, is the fact she must sign that waiver. Full stop.