From “ULTRAPRODUCTS AND ELEMENTARY CLASSES” by Keisler:
The definition of ultraproduct, and more generally of reduced product, which we shall adopt here was first given by FRAYNE, SCOTT, and TARSKI in [14], and was suggested by the "weak direct products" of CHANG and MOREL in [6], as well as by the work of Los.
From this phrase I was expecting that "weak direct products" would be an instance of ultraproducts, but then I took my definition of the former and it did not match the later:
Weak direct power
Let $e$ be a constant denoting an element in the base structure $\mathcal{M}$. The weak direct power structure $\Pi^*$ over $I$ has domain: $$ M^I_* = \{ f: I \to M \mid f(i) \neq e \text{ for only finitely many } i \in I \} $$ and interprets the symbols of $L$ as in the power structure.
Ultrapower
Let $\mathcal{M}$ be a structure, $I$ an index set and $\mathcal{U}$ an ultrafilter on $I$. The ultrapower structure $\mathcal{M}^I/\mathcal{U}$ has domain: $$ M^I/\mathcal{U} = \{[a] | a \in M^I \} $$ where $[\cdot]$ is the equivalence class under the relation $a \sim b \iff \{i \in I | a_i = b_i\} \in \mathcal{U}$. The symbols of $L$ are interpreted as: $$ R^{M^I/\mathcal{U}}([a_1],\ldots,[a_n]) \iff \{i \in I | R^M(a_1(i),\ldots,a_n(i))\} \in \mathcal{U} $$
But then the carrier of the weak direct power is merely an equivalence class, i.e. an element of the ultrapower.
Let $\mathcal{U}$ be the set of cofinite sets of $I$.
\begin{align*} M^I_* &= \{ f: I \to M \mid f(i) \neq e \text{ for only finitely many } i \in I \} \\ &= \{ f: I \to M \mid \{i \in I \mid f(i) = e(i)\} \in \mathcal{U} \} \\ &= \{ f: I \to M \mid f \sim e \} \\ &= [e] \end{align*}
Question
How is the weak direct power an instance of the ultrapower structure?
Nothing in the quoted phrase gives the impression that ultraproducts would constitute a generalization of the concept of weak direct product. It only says that reading the work of Chang and Morel would help one come up with the idea of reduced products.
Anyway, let me show that weak direct products are most definitely not a special case of ultraproducts. Chang and Morel define the weak direct product $\mathbf{M}/\sim$ of an $I$-indexed family of model-theoretic structures $M_i$ (each of the same signature) by quotienting the direct product
$$ \mathbf{M} = \prod_{i\in I} M_i $$
with the relation $f \sim g$ which holds precisely if the set $\{i \in I \:|\: f(i) = g(i)\}$ is cofinite. This is very clearly stated on page 150 of On Closure Under Direct Product. I have no idea where you got your (substantitally different) definition from: it does not appear in the Chang-Morel paper.
The reduced product generalizes the weak direct products of Chang and Morel by replacing "is cofinite" with "belongs to a fixed filter $\mathcal{F}$". Since the cofinite sets form a filter, the reduced product does generalize the weak direct product. Choosing the filter $\mathcal{F}$ as an ultrafilter gives us ultraproducts.
The importance of the Chang-Morel weak direct product construction lies in the fact that whenever a Horn formula holds in all of the factors, it holds in the weak direct product as well.
Similarly, by Łoś's theorem, if an arbitrary first-order formula holds in all of the factors, it holds in the ultraproduct.
It immediately follows from Łoś's theorem that any ultraproduct of two-element Boolean algebras is itself a two-element Boolean algebra. But the three sequences below constitute three different elements of a countable weak direct product of two-element Boolean algebras:
$$e_1 = [0,0,0,0,0,0,\dots]$$
$$e_2 = [1,1,1,1,1,1,\dots]$$
$$e_3 = [0,1,0,1,0,1,\dots]$$
Hence, weak direct powers (a fortiori, weak direct products) are not a special case of ultrapowers (ultraproducts).