My question is about what I think is a mistake in Roman Kossak's book Model theory for beginners (which is in my opinion a good book; I at least enjoyed a lot what I've read so far).
In chapter 5, Kossak defines the type $\operatorname{tp}^{\mathfrak M}(\bar a)$ of a tuple $\bar a \in M^{< \omega}$ in the domain $M$ of a structure $\mathfrak M$. He then goes on to define the notion of an isolated type. As far as I can tell, his definitions are standard.
My problem is with Proposition 5.7, which I quote in full:
Proposition 5.7. If $f$ is an automorphism of $\mathfrak M$, $f(\bar a) = \bar b$, and $\bar a \neq \bar b$, then $\operatorname{tp}^{\mathfrak M}(a)$ is not isolated.
Kossak doesn't really give a proof of that, but claims that "it follows directly from definitions and Proposition 5.3." (which only says that, is $f$ is an isomorphism, then $\operatorname{tp}^{\mathfrak M}(\bar a) = \operatorname{tp}^{\mathfrak N}(f(\bar a))$, and that much is clear). Usually, when he does that, the omitted proof is really obvious.
I think that, as stated, Proposition 5.7 is false and some of the examples Kossak gives on the next pages show that. For instance, in the ordered set $(\mathbb Z, <)$, there is only one $1$-type (so it is trivially isolated), and all elements are equivalent under the automorphism group.
I was not able to reconstruct what Kossak really meant. So my questions is: how to correct Proposition 5.7?
The correct statement is that if $f$ is an automorphism of $M$, $f(\overline{a}) = \overline{b}$, and $\overline{a}\neq \overline{b}$, then $\overline{a}$ is not definable.
We say that $\overline{a}$ is definable if there is a formula $\varphi(\overline{x})$ such that $\overline{a}$ is the unique tuple from $M$ such that $M\models \varphi(\overline{a})$. Equivalently, $\{\overline{a}\}\subseteq M^n$ is a definable set.
Note that if $\overline{a}$ is definable, then $\mathrm{tp}^M(\overline{a})$ is isolated (by the defining formula $\varphi(\overline{x})$). But the converse is not true, as examples like $(\mathbb{Z},<)$ demonstrate. Here there is only one $1$-type, which is trivially isolated, but there are no definable elements.