Have you played Magic: The Gathering before? They have a rule that any creature with any type's title also counts as they type. So if this were M:TG, Beast Warrior would be considered a Beast and a Warrior.
Not here.
EVERY Type is different in Yu-Gi-Oh!. That means Beast, Beast-Warrior, and Winged Beast are all different, even though they all contain the word "Beast." On that same token, Beast-Warrior and Warrior are considered different Types even though they both contain the word "Warrior."