In Model Theory by Chang-Keisler (2012), $\omega$-homogeneity is defined as follows (p113).
A model $\frak{A}$ is $\omega$-homogeneous if for any pair of tuples $a_1,\cdots,a_n$ and $b_1,\cdots,b_n$ in $A$ (universe of $\frak{A}$) that satisfy:
$$
({\frak A},a_1,\cdots,a_n)\equiv(\mathfrak{A},b_1,\cdots,b_n)\tag1
$$
and for any $c\in A$, there exists a $d\in A$ that
$$
(\mathfrak{A},a_1,\cdots,a_n,c)\equiv(\mathfrak{A},b_1,\cdots,b_n,d)\tag2
$$
I believe what $(1)$ means is like: for any pair of tuples $a_1,\cdots,a_n$ and $b_1,\cdots,b_n$ in $A$ and any formula $\varphi$, if
$$
\mathfrak{A}\vDash \varphi(a_1,\cdots,a_n)\iff\mathfrak{A}\vDash \varphi(b_1,\cdots,b_n)
$$
Then for any $c\in A$, there exists a $d\in A$ that
$$
\mathfrak{A}\vDash \varphi(a_1,\cdots,a_n,c)\iff\mathfrak{A}\vDash \varphi(b_1,\cdots,b_n,d)
$$
This is slightly different from the definition of elementary submodel, i.e. $\frak{A}\prec\frak{B}$ if and only if $\frak{A}\subset \frak{B}$ and for any $a_1,\cdots,a_n\in A$ and any formula $\varphi$
$$
\mathfrak{A}\vDash \varphi(a_1,\cdots,a_n)\iff\mathfrak{B}\vDash \varphi(a_1,\cdots,a_n)
$$
I ask this because I could not find the above online.
Best Answer
On the notation $$(\mathfrak{A},a_1,\dots,a_n)\equiv (\mathfrak{A},b_1,\dots,b_n):$$ Here $(\mathfrak{A},a_1,\dots,a_n)$ is the structure $\mathfrak{A}$ expanded by $n$ new constant symbols $c_1,\dots,c_n$, where the interpretation of $c_i$ is $a_i$. Similarly, in $(\mathfrak{A},b_1,\dots,b_n)$, the interpretation of $c_i$ is $b_i$. Then $(\mathfrak{A},a_1,\dots,a_n)\equiv (\mathfrak{A},b_1,\dots,b_n)$ just says that these two structures are elementarily equivalent, i.e., they satisfy the same sentences in the language $L\cup \{c_1,\dots,c_n\}$. Unpacking the meaning of this, it's easy to see that the following are equivalent: \begin{align*} (1) & \quad(\mathfrak{A},a_1,\dots,a_n)\equiv (\mathfrak{A},b_1,\dots,b_n).\\ (2) & \quad \text{tp}_L(a_1,\dots,a_n) = \text{tp}_L(b_1,\dots,b_n).\\ (3) & \quad \text{The map $f\colon \{a_1,\dots,a_n\}\to \{b_1,\dots,b_n\}$ given by $f(a_i) = b_i$ is partial elementary}.\\ (4) & \quad \text{For every $L$-formula $\varphi(x_1,\dots,x_n)$, } \mathfrak{A}\models \varphi(a_1,\dots,a_n) \text{ if and only if } \mathfrak{A}\models \varphi(b_1,\dots,b_n). \end{align*}
On the definition of $\omega$-homogeneous: The comment of Andreas Blass points out correctly that your interpretation is (still) wrong. I suspect this is a language issue (either reading or writing mathematical English) rather than a mathematical one. Note that (1) is a hypothesis in the definition. The definition could be restated as follows:
In other words, if you start with two tuples which satisfy all the same formulas, and you extend the first tuple by any new element, you can also extend the second tuple by a new element, in such a way that the resulting tuples also satisfy all the same formulas.
On the other hand, what you wrote is equivalent to "for any pair of tuples $a_1,\dots,a_n$ and $b_1,\dots,b_n$, $\text{tp}(a_1,\dots,a_n) = \text{tp}(b_1,\dots,b_n)$" which is much stronger than $\omega$-homogeneity. In fact, it's so strong as to be nonsense: it only holds in structures of size $0$ and $1$! If $|A| \geq 2$, then picking $a_1\neq a_2$ and $b_1 = b_2$, we have $\text{tp}(a_1,a_2)\neq \text{tp}(b_1,b_2)$.