Denumerably many isomorphism types
They are not trying to cheat you. You need to read the text of Example 12.12, following the formula.  It says:


  A denumerable model of $\Gamma_d$ will consist of a denumerable set $X$ with an equivalence relation in which any two elements $a$ and $b$ that are not isolated, that is, that are such that each is equivalent to something other than itself, are equivalent to each other. 


Thus $x$ and $y$ in your example must be equivalent in the model, and the picture is not an accurate representation of what $\Gamma_d$ captures.