On an (possibly) equivalent way of expressing closure under limits for a full subcategory

Question

I will refer to the notions recalled in

On Proposition $13.27$ of Adamek: full reflective subcategories are limits closed iff they are replete?

where I asked for clarifications about a result stated in "The Joy of Cats". The question I want to pose is the following: let $A$ be a (non necessarily replete) full subcategory of $B$. Is it true that the property of being closed under the formation of products and equalizers (or of products, pullbacks and terminal objects) is equivalent to that of being closed under the formation of all limits?