It seems to depend on the country.
"el uno de junio" is used in Spain, "el primero de junio" in Latin America.
I'm on a one-way street: Isn't there anybody that could decide from all the possibilities how the date should be acceptable both in English and Spanish? I only get a feeling that there possibly is someone or something making a simple thing harder than it is - for what? ( to get funny with fumbling students?)
The difference is a verb form versus a noun.
Noun (English and Spanish)
Empiezas el trabajo el uno de junio = You start work on June first.
Infinitive (Spanish) and Gerund (English)
Empiezas a trabajar en junio primero = You start working on June first
Notice that empezar is a verb that uses the pattern
Verbs + a + infinitive
Empiezas a trabajar