A kernel of a mapping $h$ from $\mathfrak{A}$ to $\mathfrak{B}$, generally, is an equivalence relation $\{(a,a') \in \mathfrak{A} \times \mathfrak{A} \mid h(a)=h(a')\}$.
However, in model theory, there is another version of definition of kernel of mappings. That is, $\ker(h) \colon=\{\varphi(\bar a) \mid \models_{\mathfrak{B}}\varphi[h(\bar a)]\}$, in which $\bar a$ is a non-repetitive enumeration of $\mathrm{dom}(\mathfrak{A})$, $\varphi(\bar a)$ are all atomic sentences.
Let's call the former one set theoric version, the later one model theoric version.
My question: What's relationship between them? Are they equivalent to each other in some senses?
Elaboration for the model theoric version
The model theoric version I found was in exercise 1.5.4 in Hodges's Model Theory. This exercise mainly requires a proof of those assertion following are equivalent to each other:
(a) $\mathrm{diag}^+(\mathfrak{A}) \subseteq T$ and $T$ is =-closed;
(b) $T$ is the kernel of some homomorphism from $\mathfrak{A}$ into $\mathfrak{B}$;
(c) $T$ is the kernel of some surjective homomorphism from $\mathfrak{A}$ into $\mathfrak{B}$;
In which $T$ is a set of atomic sentences of $L(\bar a)$.
Since I haven't found an explanation why he defined kernel in this way, I asked this question here.
If the $ker_m(h)=diag(A)$, then the $h$ is an embedding and viceversa(see diagram lemma in hodges book). Hence, $ker_m(h)\neq diag(A)$ iff the set-kernel has a equivalence-class with more then 1 element.