Firebase AuthenticationユーザーのdisplayNameは、Firestore rules上ではnameとして定義される

import * as Firebase from 'firebase'

const auth = Firebase.auth()
const email = someEmail
const password = somePassword

try {
  const credential = await auth.createUserWithEmailAndPassword(email, password)
  const user = credential.user
  if (user === null) {
    throw new Error(
      `creating user for authentication was success, but nobody is logging in. something wrong! credential: ${credential}`
    )
  }

  await user.updateProfile({ displayName: 'Haskell' })  // !
  await user.sendEmailVerification()
} catch (e) {
  if (e instanceof Error && /The email address is already in use by another account./.test(e.message)) {
    throw new Error(`メールアドレス「${account.email}」は既に使用されています。`)
  }
  throw new Error(e)
}

const ref = await Firebase.firestore()
  .collection('accounts')
  .doc('Haskell')  // !
const doc = await ref.get()

if (doc.exists) {
  throw new Error(`アカウント名「${account.name}」は既に使用されています。`)
}

ref.set(account)

const ref = await Firebase.firestore()
  .collection('accounts')
  .doc('Haskell')  // !
const doc = ref.get()

rules_version = '2';

service cloud.firestore {
    match /databases/{database}/documents {
        match /accounts/{account} {
            allow create;
            allow read: if account == request.auth.token.name;
            // 誤り ↓
            // allow read: if account == request.auth.token.displayName;
        }
    }
}