Definition of $\omega$-homogeneous and elementary equivalence 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.
 A: *

*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:

A structure $\mathfrak{A}$ is $\omega$-homogeneous when: for any pair of tuples $a_1,\dots,a_n$ and $b_1,\dots,b_n$ from $A$, if $\text{tp}(a_1,\dots,a_n) = \text{tp}(b_1,\dots,b_n)$, then for all $c\in A$ there exists $d\in A$ such that $\text{tp}(a_1,\dots,a_n,c) = \text{tp}(b_1,\dots,b_n,d)$

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 $\text{tp}(a_1,a_2)\neq \text{tp}(b_1,b_2)$.


*On sources: You write (in the comments) "I can not find above definition (and interpretation) online". I expect you can find a definition of $\omega$-homogeneous (or $\kappa$-homogeneous, more generally) in any introductory model theory textbook (in addition to Chang-Keisler, the standard books are by Marker, Hodges, and Tent-Ziegler). As with most technical areas of mathematics, it's much easier to find complete information in textbooks than online. But with a quick Google search, I found the definition of $\kappa$-homogeneous model here and here (Definition 4.4.1 on p. 26).

