She sounds like a nice client. She hasn't asked you for a refund, just advised you that she's cancelling her next appointment as she has found the roots come through too quickly after having the service with you vs. her previous service provider.
I personally think it's amazing that the first thing you did was reflect on what you could have done to make the service better and what possible improvements you could have made to your techniques.
When you say you are newly qualified, is that level 2 or 3? And was that studied in a salon or at full time college?
Your attitude is amazing, and I'd hire you if you were in the North West. I have a team of 13 people in secure well paid jobs, and we're expanding all of the time. The one thing I find it difficult to do is drum into people the difference between constructive criticism and a moaning client. You've just had the former, and I have a feeling that you know (and understand) that fact.
Maybe she's just a client who's alarmed at how much root regrowth has come through after less than 21 days? Maybe her previous foiler was better at getting closer to the root? Maybe she has loads of friends and loads of money, and lots and lots of influence?
We had a (NEW) client this week tell us that name badges would be nice as she didn't know everyone's names. The name badges are currently on their way and being engraved as we speak.
In a recession you can't afford to ignore your clients' needs. If you do, people like me will come and snatch them from you.