The normal closure $N_R$ of a subset $R$ in a group $G$ is the subgroup generated by $\bar R=\{g^{-1}rg|g\in G,r\in R\}$.
I read it is the smallest normal subgroup such that every element of $R$ is identified with the identity.
What I see is that for every $r\in R$ we have $[r]=[e]$ since $r\cdot e^{-1}=e^{-1}re\in N_R$ (or $rN_R=N_R$), which means $r\sim e$.
Is this what "every element of $R$ is identified with the identity" means? Equivalent in the equivalence relation of cosets of $N_R$?
Should I get used to $=$ meaning $\sim$ in the case of relations as written in presentations of groups?
The author means that $N_R$ is the smallest normal subgroup such that elements of $R$ are identified with the identity when you quotient by $N_R$.
This is no different from saying $N_R$ is the smallest normal subgroup which contains $R$.
Concerning you're last question, I think you have the right idea. When we impose the relation that some expression is equal to the identity, this just means that we are taking the quotient by the smallest normal subgroup containing that expression.