She’s really, really, really black

My 3-year-old daughter will turn 4 in a week. She’s Amerasian. I want to raise a racially aware child because she will likely present as white, and I want her to be aware that a lot of people, including her father who has unusually dark skin by Chinese standards, face a lot of rude behavior from people just because of skin color. As a part of this educational process, we watched the movie Harriet about a month ago. https://www.youtube.com/watch?v=GqoEs4cG6Uw
From the movie about the life of Harriet Tubman, Ann learned that people with dark brown skin can be referred to as black. The week after watching the movie, she said to me as we started one of her many classes, referring to a colleague’s mom, “She’s black like Harriet.” Yes, she is, I replied. I wanted to discuss more, but all parents were wrangling toddlers out of jackets and into chairs and the teacher was there and the class was starting before I could get more conversation either with my daughter or with the mom she had just compared with the stunning beautiful actress who portrayed Harriet in the movie.
Today, we were leaving her pre school at noon on our way to pony riding class. I encouraged her to wish her favorite teacher, Ms. Jones to have a wonderful evening, and she did with unusual enthusiasm. Ms. Jones responded equally affectionately, rubbing both of her checks and wishing Ann a wonderful evening. As we walked forward, Ann said to me, “She’s really, really, really black.” Yes, Ms. Jones is likely African American, although as with most American citizens, she’s likely to have more than one racial/ethnic category. However, since the first comment, I wanted to get into this conversation more deeply and since we weren’t pressed for time, I stopped right there and said, “I’m also white. And you are Asian.” Ann did a double-take. “What is Asian?” Actually, you are not Asian, I explained. Your dad is Asian. Your grandma is Asian. But you … Ann cut me off. “What is Asian?” she asked. Asian is people like your father from places in Asia like China. “I’m Asian?” She asked. Well, you’re half Asian. You got Asian genes from your dad and white genes from your mom. Then she said I want ice cream.
I’m posting this because I have no idea if I’m doing this right or wrong. Kids don’t come with manuals, and I have no idea to how to approach conversations like this.
We had another encounter completely independent but related to the idea of accepting the people in your neighborhood. We were leaving ice skating class maybe 2 or 3 months ago and a group of women and girls dressed in burkas came into the ice skating rink. Ann stopped and stared at them for a long time, and I was unable to continue the clothing changeover and backpack organization that goes with leaving the rink because she wouldn’t move. She just stared at the group from a distance of a meter or two.
I walked her over to the group and asked Ann to give a salutation in Arabic from the Qu’ran. I forgot which one I selected. But she was unable to pronounce it, so I said that’s okay, you can say Hello.
“You won’t understand this right now, but your mom is Christian. Your grandma is Buddhist. You dad generally talks like a Taoist and your new friends are Muslim. These are people in our neighborhood. Let’s say hello to them.”
By this time, the girls who had been getting shoes off and skates on stopped and looked at us. We greeted them. They said hello. The oldest woman in the group, possibly grandma for one of the girls thanked me. After the hello, Ann didn’t stare any more and women in burka seemed to have achieved a non-alien status.

I’m happy for any inputs on these topics, including criticism, because I’m not sure if I’m doing it right, and I’d like to get it right. All ideas welcome.