Dependent Sums and Products
I think what's confusing you is that $A \times B$ is both a product and a coproduct:


It is the product of two factors, namely $A$ and $B$.
It is the coproduct of $A$-many copies of $B$.


Once you realize this, you will see that we can obtain $A \times B$ as both a $\sum$ and a $\prod$:


Take $P : \mathtt{bool} \to \mathsf{Type}$ where $P(\mathtt{false}) = A$ and $P(\mathtt{true}) = B$. Then
$$\sum_{b : \mathtt{bool}} P(b) \simeq A + B$$
and
$$\prod_{b : \mathtt{bool}} P(b) \simeq A \times B$$
Take $Q : A \to \mathsf{Type}$ where $P(x) = B$ for all $x : A$. Then
$$\sum_{x : A} Q(x) \simeq A \times B$$
and
$$\prod_{x : A} Q(x) \simeq (A \to B)$$


We should therefore not pay attention to $A \times B$ when deciding on a good naming scheme for these two constructs.

The dependent exponential is exactly a dependent product.