I am self-studying dynamical systems, and wanted to double-check my understanding of these concepts.
In Strogatz's "Nonlinear Dynamics," the author plots a periodic solution to the forced double-well oscillator problem, then plots a chaotic solution to the same system. Without writing out the whole equation, we fix the forcing term as $$ F \cos t, $$ so it has period $T=2\pi$. In an effort to get more insight into the chaotic nature of the solution, he defines a "Poincare section" as the collection of points $$\left\{ \left(x\left(t\right), y\left(t\right)\right) \, \middle| \, t = 2n\pi, n\in \mathbb{N}\right\}$$ which I'm interpreting as "where the solution is at equal intervals in time." He chose the period of the section as equal to the period of the forcing, though no formal reasoning is given.
This website defines "Poincare sections" similarly, and the picture of one looks a lot like what you obtain for Strogatz's example.
Where I'm getting confused is that the definition of "Poincare map" seems to be very related to "Poincare section". The Poincare map is a function, defined as the point in a specific subset --- a "section" --- that a trajectory returns to, not necessarily at equal intervals of time. Thus, the concept of "Poincare section" and "Poincare map" are distinct, since one is defined by space and one by time. In the forced double-well example, we were unambiguously discussing a section, since we considered equal intervals in time, and with chaotic trajectories there was no guarantee that a trajectory would return to the same "section" after a given interval of time.
However, on the wikipedia page for Poincare map, the image appears to be a Poincare section, as I defined above, not a Poincare map. I'm guessing this is a result of someone else's confusion, or else such a section is somehow a visualization of a Poincare map (though I highly doubt it). This is making me doubt my understanding; am I correct in assessing the difference between these concepts?