Thanks to the answer here, I think I understood what are elementary/atomic diagrams.
I'm reading Hodges' textbook now, and he defines a diagram a bit differently: for $A$ an $L$-structure and a sequence $a=(a_1,a_2,\dots)$ of elements of $A$, he adds to the signature $L$ the sequence of $c=(c_1,c_2,\dots)$ of distinct constant symbols and interprets them as usual. He then defines the diagram almost in the same way as Marker (the set of closed literals in the new signature which are true in the new structure) but he also requires that $A$ be generated by the set of all elements in $a$. What is this requirement for? Is this definition more general? (In Marker's context I think this holds automatically because he assigns a new constant to every element of $A$).
The definitions amount to the same thing. Hodges allows the language $L(\bar a)$ for the diagram to have fewer constant symbols than there are elements of $A$, but compensates for it by requiring that the interpretations of the symbols at least generate $A$. The important thing here is that the diagram fully determines the truth value of every quantifier-free sentence of the full language $L(A).$ This is clearly true in Marker's definition: just break it down to the atomic sentences and determine their truth values by whether or not they're in the diagram. In Hodges' definition we need to do a little more work and first express any constant from $L(A)$ that is not one of the $a_i$ in terms of the $a_i.$ In order to do this, we need to assume that the $a_i$ generate $A.$