Her paper that you link to is strongly related to the 2nd half of the lecture series. Delayed embedding is used to identify chaotic attractors, stable and unstable fixed points, unstable periodic orbits (with a spectrum of periods), etc. Unfortunately, visual inspection of delayed embedding for lotto ball picking systems has non of these structures that can be visually identified in embedded space. It looks like noise.
2nd order statistics (pairs) covers any information you can pull from a 2D delay embedded space. Same is true for higher order statistics and higher dimension embedded spaces.
The phase space structure of the original system is preserved in the corresponding delayed embedded space, with the caveats that you sample fine enough, long enough, not too fine, and with noise minimized or eliminated, so as to capture the geometry in phase space. If you can identify geometries in the delay embedded space (with the sampling caveats satisfied), then you know they are also in the system being analyzed (nice!). You can then use those geometries to predict future system output, control the system, etc.