You have axiomatized the theory asserting that $E$ is an equivalence relation with infinitely many classes, ALL of which are infinite. Your statement $\psi_n$ says that every class has at least $n$ elements.
But the theory mentioned in the title problem was just that $E$ should have infinitely many infinite classes (and perhaps also some finite classes). It turns out that this problem is impossible.
Theorem. The collection of equivalence relations with infinitely many infinite classes is not first-order axiomatizable in the language of one binary relation with equality.
Proof. Let $E_0$ be an equivalence relation on a set $X$ with infinitely many classes of arbitrarily large finite size, and no infinite classes at all. So $\langle X,E_0\rangle$ is not one of the desired models. Let $T$ be the elementary diagram of $\langle X,E_0\rangle$, plus the assertions with infinitely many new constants $c^n_i$, that $c^n_i\mathrel{E} c^n_j$ and $c^n_i\neq c^n_j$ and $\neg (c^n_i\mathrel{E} c^m_j)$, whenever $n\neq m$ and $i\neq j$. This theory is finitely consistent, and so it is consistent. Any model of $T$ will be an expansion of an elementary extension of the original model $\langle X,E_0\rangle$, but it will now have infinitely many infinite classes.
Thus, the property of having infinitely many infinite classes is not first-order expressible, since the original model did not have that property but the new model did, even though they were elementary equivalent. QED
Here are two examples :
$1 - $ Consider the relation $\equiv$ ( an equivalent relation), then
$$a \sim b \Leftrightarrow a\equiv b \mod 2 $$
That is, $a$ and $b$ will be in the same class $\overline{a}$ if their remainders of the division by $2$ are the same. For example $4$ and $6$ belong to the same class, which we are going to choose a representant $0$, because
$$6 = 3 \cdot 2 + \color{red}{0} \ \ \text{and} \ \ 4 = 2 \cdot 2 + \color{red}{0}$$
then we say $\overline{4} = \overline{6} = \overline{0}$. If we think, there are two distinct classes: $$\overline{0} = \{x \in \mathbb Z ; x \equiv 0 \mod 2, \text{$x$ is even}\}\ \ \text{and}\ \ \overline{1} = \{x \in \mathbb Z ; x \equiv 1 \mod 2, \text{$x$ is odd}\}$$
The set of all classes is
$$\mathbb Z_2 = \{\overline{0}, \overline{1}\}$$
$2-$ Consider the relation
$$(a,b) \sim (c,d) \Leftrightarrow ac = bd $$
This equivalent relation gives us the fractions, that is the filed of fractions of $\mathbb Z$. Similarly we choose a class representant for example,
$$\frac{1}{2} = \frac{2}{4} = \frac{3}{6 } = \cdots$$
we choose $\frac{1}{2}$ to be the class representant. Notice that $\mathbb Q = \{ \frac{a}{b} ; a,b \in \mathbb Z, \ \ \text{where}\ \ b \neq 0\}$ is the set of all classes.
Best Answer
You are right that $\mathcal{C}_{< \omega}$ is not axiomatisable. The usual argument using compactness goes by contradiction. It goes as follows (I'll leave it to you to fill in the details).