On the correct definition of attractors It is well-known in dynamical systems that the concept of "attractor" differs in the literature.
My question is whether attractors need to be defined as subsets of $\omega$-limit sets of some point on the phase space, or can be any compact set in the phase space? (What important results don't hold for the more general definition, which important example are (not) covered by the more restricted definition etc.?)
On one hand, in the link above, as well as in the well-known book "Handbook of Dynamical Systems Vol. 1A" by Hasselblatt & Katok, which are well-known authors and researchers, an attractor there can be any compact set.
On the other hand, in the book "Dynamical Systems and Chaos" by H. Broer, F. Takens, which are equally well-known names in the field, attractors are only defined as special types of $\omega$-limit sets.
Unfortunately, I know no reference that unifies these definitions and clarifies their relationship.
 A: This is just an extended comment, as I'm not even trying to recall all the uses of the term attractor in the literature, pointing out when equivalence holds or when there are some subtle differences depending, for instance, on the topology of the phase space or on properties of the map. Even the more basic definition of $\omega$-limit set of a certain subset $Y$ of the phase space is not defined completely uniformly in the literature when $Y$ is not just a finite set of points.
I just want to point out what in my view is a very clean, purely topological definition, from which it follows that attractors coincide with $\omega$-limits of inward sets. This is the approach I like best, since it naturally comes from the concept of stable, attracting set. Attractors are closed attracting sets which are strongly invariant, that is they coincide with their image.
Definition: Let $X$ be a compact metric space and $f$ a continuous map. An attractor is a nonempty, closed set $Y\subset X$ such that $f(Y)=Y$ and, for every $\epsilon>0$, there is $\delta$ such that all points $x$ such that $d(x,Y)<\delta$ will stay within distance $\epsilon$ (that is $d(f^n(x),Y)<\epsilon$ for every $n$) and verify $d(f^n(x),Y)\to 0$ when $n\to\infty$.
Assuming this definition you can easily prove the following:
Fact: Attractors are the $\omega$-limits of (nonempty) inward sets, that is of sets $Z\subset X$ such that $f(\overline{Z})\subseteq Z^\circ$.
See for instance Propositions 2.64-2.65 in Kurka, P. (2003). Topological and symbolic dynamics. Société mathématique de France.
You can also see Akin, Ethan. The general topology of dynamical systems. American Mathematical Soc., 1993, p. 43, Theorem 3, which contains some equivalent definitions of preattractor and attractor in the more general case when $f$ is a closed relation on the phase space. They are consistent with the definition above and the author points out the important link with the concept of chain-recurrence (see in particular point c)-1.).
