Meaning of closed under an associative product

Question

I know the meaning of associative binary operation. I know the meaning of closed under a binary operation. Does closed under an associative product means:

$$ a*(b*c) \in G \implies (a*b)*c \in G$$

Answer 1

(closd under associative product *) = (closed under product *) + (product * is associative).

Answer 2

No. It means closed under a product, which happens to be associative.