As I'm not allowed to ask a new question due to limit reached matter,
I still want to **EDIT** this one as communicated with @Alex Kruckman
in the comments below. I would like to understand the **relationship** of the two
definitions relating AEC's notion of a **type** by assuming existence
of **amalgamation** in Jarden's approach with realizing formulas in elementary extensions in Hodge's approach. What should I take for an AEC $(K,\preceq)$
to obtain exactly Hodge's notion and prove its equivalence with the amalgamation
approach ?

**OLD Question:**

I have come across two different definitions of saturation and I would like to relate them.
One is in Wilfrid Hodges book (first snippet below) and it says that an elementary extension $B$
realizes all types with less than $<\lambda$ elements. The other approach is the definition 1.0.25 in the second snippet below.
It says that $M$ of cardinality $\lambda^+$ is saturated if $M$ realizes type of every sub-model of cardinality $\lambda$. I do not even know what is to be proved for an equivalence of these two
approaches. It appears to me that in the second approach we are looking at **smaller** models and in the first approach at **larger** elementary extensions.



[![enter image description here][1]][1]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/Hy5zN.jpg
  [2]: https://i.sstatic.net/eAkhm.jpg