How do you call it when two celestial bodies come as close to each other as they will in their current orbits?
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?
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)