I have a theory $T$ such that the class $K$ of finite models of $T$ has a Fraïssé (Fraisse) limit $L$. The theory of $L$ happens to be the model completion of $T$. The limit $L$ is of course locally finite, but it is not uniformly locally finite.
I'd like to know what the (0-) definable elements of $L$ are. I would imagine that since $L$ is ultrahomogeneous there are not many definable elements. (I actually doubt if there are any definable elements that are not represented by closed terms.) The following are my observations:
If $a \neq a'$ have the same quantifier-free type (over 0), then they are conjugate by an automorphism of $L$, so neither of them are definable.
If for $a$ there are no other element of the same quantifier-free type in $L$, the element $a$ is defined by the formula that is equivalent to that type (which exists because $L$ is locally finite).
I also had the following idea: given $a \in L$, if I can find multiple copies of $\langle a \rangle^L \in K$ in $L$, it might be helpful for me to find $a' \neq a$ with the same quantifier-free type. But I don't know if I always can find such a substructure, nor do I believe that this is always helpful for the purpose.
Can we tell more about what are definable elements in $L$ by abstract nonsense? Or do I have to get into the details of $T$, such as how the amalgamation works in $K$?
By your observations, understanding $\mathrm{dcl}(\emptyset)$ completely reduces to the following question:
Consider a pair $(A,a)$, where $A\in K$, $a\in A$, and $\langle a \rangle = A$. Can we find another structure $B$ in $K$ and two embeddings $f,g\colon A\to B$ such that $f(a)\neq g(a)$?
For every pair $(A,a)$ such that the answer to the above question is no, there is a unique embedding $f_{A,a}\colon A\to L$. The $\emptyset$-definable elements in $L$ are exactly those of the form $f_{A,a}(a)$.
For a silly example where there are $\emptyset$-definable elements which are not named by closed terms, consider the language $\{P\}$ where $P$ is a unary relation symbol and the theory $T= \{\forall x\forall y\, (P(x)\land P(y)\rightarrow x=y)\}$ which says there is at most one element satisfying $P$.