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.