Your teacher is right, but you are confusing "more commonly" with "exclusively". Here's an explanation:
Gallina is the female (hen)
Gallo is the male (cock/rooster)
Pulcino is the newborn (chick)
Galleto is a young male ('teen' / not yet a full rooster)
Chioccia is while the hen is brooding or is taking care of its chicks
Pollo is for the species name (chicken) or for males reaching maturity (which is the moment when they are slaughtered thus leading to "chicken meat" – carne di pollo – in the supermarket), for females reaching maturity it's pollastra (though you only use this if you really want to make such distinction).
And it's common to say gallina to call the species instead of pollo when they are alive because it's always dozens of hens with a single rooster.