What's her name again?

When I dropped Ember off at Preschool yesterday there was a relief teacher we hadn't met before.  Ember was quite taken with her, partly because she had long hair, and Ember (being obsessed with Rapunzel) is desperate to grow hers long too.  I said hello to the teacher, she introduced herself as Pei Chen, and Ember (who demands that I "give [her] to a teacher" before I'm allowed to leave) was happy to give Pei Chen a cuddle as I was leaving.

That evening when we were talking about Ember's day, I said to her:

Me: Did you like the new teacher today?
Em: What teacher?
Me: Pei Chen, the teacher you were with this morning, with the beautiful long hair.
Em: Oh, yes I did. (pause) But Mummy, is it ok if I don't like a teacher's name?
Me: You mean the name Pei Chen?
Em: Mmmm.
Me: It's ok if you don't like someone's name, but you probably shouldn't say it to them.  Her name sounds a bit different to you because it's Chinese.
Em: What is?
Me: Her name is Chinese.
Em: Her name is Chinese?  I thought it was Pei Chen!
Me: (struggling not to laugh) No, her name is Pei Chen, but it's Chinese.  From China.
Em: So I can call her Chinese?
Me: No, because that's not her name.  Her name is Pei Chen.
Em: Then why did you say her name is Chinese?
Me: Because it's from the country China.  Your name, and names like Tamsyn and Cate and Connor are New Zealand names.  If you were from China, your name might be something like Pei Chen or Mei Ling or Xiaojing, and you would probably think "Ember" sounded strange.  Or if you were from Japan you might be called Kaoru or Michiko or Akiko.
Em: Yoshi [another Preschool teacher] is from Japan!
Me: Yes, and Yoshi's name is Japanese.
Em: Oh, I thought it was Yoshi...

I gave up after that!