Regarding the second question.

In one case you focus mainly on ODE, and in the other

You focus on dynamics.

"dynamical systems" is more commonly used in mathematics and science to specifically describe systems governed by differential equations

In mathematics, the Dynamical System is the study of the long term evolution of a system that is expressed by one of two means. A continuous dynamical system is determined by an ordinary differential equation, i.e., a vector field in the ambient space be in a subset of R^n, C^n or a manifold. The discrete dynamical systems is a system expressed a difference equation or more generally, interactions of a mapping on a base space, again a subset of R^n, C^n or a manifold. Historically these systems might have raised out of questions in physics, statistical mechanics and other sciences but now encompass a major area of mathematics. There is interplay between continuous dynamical systems and discrete dynamical systems. For example in studying the periodic structure of solutions of ODE, one powerful took is to construct the first recurrent map, a.k.a., the Poincaré map. There is major tie ins with probability theory, ergodic theory and many other areas of mathematics. The Poincaré map.is a discrete dynamical systems of one fewer dimensions than the related continuous dynamical system. https://en.wikipedia.org/wiki/Poincaré map.

The following reference is a good starting point.

The richness of the long term behavior of dynamics is in at the stationary points of the systems. In case of the ODE, x'=f(x) it is the behavior at equilibrium of the vector field f, i.e. f(x0) = 0 or the equilibrium. In the case of a discrete dynamical systems it is at the fixed points of the defining map,

T: M->M, that is T(x0)=x0.

https://www.amazon.com/dp/0387971416?asin=0387971416&revisionId=&format=4&depth=1