If you are applying for an internship, you should probably apply for the internship FIRST, before you have a reference letter sent.
Especially if we didn’t request reference letters, but names of references.
And you should also give your references enough information that they can say what you are applying for in their letter.
That way, I don’t have to send your reference letter to everyone, asking “Do you know who this kid is? Are you hiring them?
It’s not a good first impression.