Definition of ergodic map

Question

I ask the similar question before. About definition of Ergodic theorem. Now just sincerely ask another fundamental problem about the definition of ergodic map.

The following definition is what I read in my textbook:

Let $(X,\mathcal{A},\mu)$ be a probability space. Let $T: X\rightarrow X$ is $\mu$-invariant ($\mu$-preserving). Then $T$ is ergodic if for every $E\in \mathcal{A}$ with $T^{-1}(E) = E$ if and only if $\mu(E)=0$ or $1$.

In this definition, it requires $T^{-1}(E) = E$. However, in the following paper, it seems to use $T(E) = E$ as the definition.

https://www.tandfonline.com/doi/abs/10.1080/10236190601045788 (p.1154, Def 2.16, please also see "$A$ is invariant, i.e. $T(A)=A$ on the top of that page.")

My questions are:

1. What is the difference between them? or is the definition in the paper not correct?
2. For the definition by using $T^{-1}(E) = E$, does it imply $E$ is not an attractor? since the only initial condition such that $T(x_0) \in E$ is $x_0 \in E$ instead of $x_0\in X\setminus E$.

So if we want to discuss $E$ being an attractor, we have to use $T(E) = E$ since this could imply the possibility that $T(x)\in E$ for $x\in X\setminus E$. Do I make sense?

Thanks in advance.