Finitely generated substructures and model completions

Finitely generated substructures and/or local finiteness seem to play a great role in the study of model completions. For instance, I am told the following:

A universal theory $T$ has a model completion $T^*$ iff every e.c. model of $T$ satisfies $T^*$ and for every $M\models T^*$, $M_0 \subseteq M$ finitely generated, and $M_0 \subseteq M_1 \models T$, there is an embedding from $M_1$ to an elementary extension of $M$ that preserves $M_0$ pointwise.

While it is easy to prove this, I wonder if there is anything about finitely generated substructures and model completions. Is there a good source from which I can learn similar facts? Obvious sources like the relevant chapter in the Handbook of Mathematical Logic, Chang & Keisler, the larger Hodges, or Marker was not helpful.

• "e.c." is ? equationnally compact ? then is $T$ only in a functional language ?(that's what the "universal algebra" tag suggests but...) – Maxime Ramzi Dec 4 '17 at 22:00
• @Max existentially closed – Alex Kruckman Dec 4 '17 at 23:37
• @AlexKruckman : Thanks, I was completely mistaken ! – Maxime Ramzi Dec 5 '17 at 7:11