Your question is not specific to the evaluation map.
Consider any morphism $f : A \otimes B \to C$ in a in a symmetric monoidal (not "monodical") category. There are two adjuncts because $A \otimes B \cong B \otimes A$:
- We can take $A$ to the other side of the arrow to get $f_1 : B \to [A, C]$.
- We can take $B$ to the other side of the arrow to get $f_2 : A \to [B, C]$.
Now let us apply this to the evaluation map $e : [X, Y] \otimes X \to Y$:
- In the first case we get $e_1 : X \to [[X,Y], Y]$.
- In the second case we get $e_2 : [X,Y] \to [X,Y]$. This is what the nLab page is talking about.
Now, if I had to guess where you made an error, I would hypothesize that you considered the first case, and additionally you confused which is the left and which is the right adjoint: while it is the case that $[Y, [X, Y]] \cong [X \otimes Y, Y]$, it is not the case that we can just switch things around and expect also that $[[X, Y], Y] \cong [Y, X \otimes Y]$. Internal hom-sets are not symmetric!