From my understanding, the Earth (Solar System) crosses a center line of the galactic plane (ecliptic *edit wrong definition, sorry!), while orbiting around the Sun. The Sun also wobbles up and down along this same center plane, while orbiting the galactic center.
Are there any connections of this rhythm to our seasons or climate here on Earth? Are there any definitions or any sort of study, reference, or name, for this concept?
See the Second and Third images on this article…
I think you are asking whether there are any consequences for the Earth of the Sun and Solar System crossing the galactic plane$^1$? Yes, there have been a number of studies suggesting that their might be an affect on the Earth in terms of an increase in cometary bombardment due to tidal forces in the Galaxy. However, there is no conclusive evidence that this has happened.
The orbit of the Sun around the galaxy is basically a circle, but superimposed on that are small, roughly sinusoidal excursions up and down and in and out. The vertical oscillation has a period of about 70 million years, which means the Sun passes through the Galactic plane every 35 million years. See for example How far is the Earth/Sun above/below the galactic plane, and is it heading toward/away from it? and What mechanism causes oscillations of the solar system's orbit about the galactic plane?
How could this affect the Earth? Well, the amplitude of the motion is about 100 pc, which is only half the scale height of the galactic disc. This will mean the local stellar density around the Sun will change a little. But since stellar close encounters are in any case rare, this is unlikely to provide a distinct modulation in any phenomena on Earth.
A better bet might be the changing tidal influence of the galaxy on the Oort cloud. Some calculations (e.g., Matese et al. 1995) have suggested there might be effects sufficient to modulate the rate of comets entering the inner solar system. This could result in impacts, mass extinctions, climate events etc. Gardner et al. (2011) calculate that the cometary flux would be modulated by about 30 per cent.
The observational evidence for such episodes is reviewed by Rampino & Prokoph (2020). They look at 58 analyses of cratering and conclude there is some evidence of a 26 million year periodicity and it is tempting to associate this with the galactic plane crossings (though the period is a bit discrepant). The problem with this interpretation is that most of the cratering isn't due to comets from the Oort cloud, and material in the inner solar system and asteroid belt would not be affected by galactic tides in the same way.
$^1$ The orbit of the Earth around the Sun has no bearing on this issue. You might be confused by the scale of the diagram in the source you link to. The Earth does not cross the galactic plane once per orbit and the size of the Earth's orbit is tiny compared with the excursions the entire Solar System makes above and below the galactic plane.
