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A model is strongly $\kappa$-homogeneous when every partial elementary map between its subsets of size less than $\kappa$ extends to an automorphism. Here the unqualified expression homogeneous model uses this automorphism extension convention. For finite parameter tuples it says, in particular, that equal complete types can be carried to one another by an automorphism.

A weaker local convention asks only for one-point extensions of small partial elementary maps. A $\kappa$-saturated model always has that local property. If its cardinality is $\kappa$, the back-and-forth method yields the strong version too. For example, order-preserving maps between finite subsets of the rational dense linear order without endpoints extend to automorphisms by back-and-forth.

For $\kappa =|N|$, saturation means that every complete type over a subset of $N$ of cardinality less than $\kappa$ is realized in $N$. The model $N$ is a universal model when every model of $T$ of cardinality less than $\kappa$ elementarily embeds into $N$, and it is a homogeneous model when every partial elementary map between subsets of $N$ of cardinality less than $\kappa$ extends to an automorphism of $N$.