To ask who is right?

[sweeneytoddsrazor]

At a friend's house earlier and she asked Alexa to play some Betty Midler. Sure enough Alexa responded shuffling songs by Betty Midler. I always thought Bette was pronounced Bet. Have I got it wrong?

No. I think that your pronunciation is correct.

I think Alexa probably just knows it's a common mistake. Like when you Google something and it says, "did you mean _?"