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On Earth, we say Mars is in opposition when it is 180º from the Sun, which also marks the times when Mars and Earth come closest to each other.
Is there a specific name for such proximity events? For example, Saturn and Jupiter come to this maximal proximity every 20 years or so. Is there a name for the event itself, the "approaching" (previous) phase and the "departing" (next) phase?
This is a syzygy. "A syzygy of the Sun, Jupiter and Saturn." In general a syzygy is when three bodies are in a line. In your case the three bodies are the sun, and the two planets. You can also talk about "Saturn is at opposition as seen from Jupiter. Not every syzygy is "proximity event" but many are. I know of no word or phrase for the "approaching phase" or the "departing phase".
I don't think that there is a specific term for "closest approach". Establishing absolute distance is generally difficult, compared with finding the direction. And the closest approach doesn't happen exactly at opposition (due to orbital tilt and eccentricity). Terms like "opposition" or "syzygy" are about the direction rather than distance.
a conjunction or opposition, especially of the moon with the sun. So I guess A conjunctive syzygy may be used to define the proximity event?
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Close approach seems to be the prevailing term for an event when two objects independently orbiting the Sun pass a minimum distance from each other. JPL and ESA use this term with near-Earth asteroids even if the approach is not especially close.
Since real orbits are not perfectly concentric or coplanar, this generally does not occur at exactly the same time as opposition. In the 2020 apparition of Mars as seen from Earth, opposition occurs on October 13, and close approach occurs on October 6. Even in the perihelic Mars apparition of 2003, opposition and close approach were 1.6 days apart.
What a linear alignment of three bodies is called depends on the observer's point of view. At the syzygy of 2020-11-02, an observer on Jupiter would see Saturn at opposition, and an observer on Saturn would see Jupiter at inferior conjunction. Their close approach is on 2020-10-12 at a distance of 4.88 au. (source: JPL HORIZONS)
perihelion.
Definition from Google
the point in the orbit of a planet, asteroid, or comet at which it is closest to the sun.
Apsis denotes either of the two extreme points (i.e., the farthest or nearest point) in the orbit of a planetary body about its primary body (or simply, "the primary").
For generic situations where the primary is not specified, the terms pericenter and apocenter are used for naming the extreme points of orbits
As @Rory notes, periapsis is also a relevant term which, according to Wikipedia, is preferred to pericenter in technical usage. Likewise with apoapsis, I would add. But Wikipedia notes these terms can also be used to refer to distances rather than points.
"An inferior conjuction occurs when the two planets lie in a line on the same side of the Sun." (From wikipedia)
I have not found any terms referring to the two different portions of the orbit of two bodies, between the nearest and furthest points.
I think the term "Appulse" is what you're looking for here. The 2020 December "Christmas star" event where Jupiter and Saturn appeared as close as they could appear for the year (minimum apparent separation), is a classic appulse between two or more objects within our field of view. The Wikipedia definition can be found at the link below (with links to the other geometries mentioned above):
