Unfortunately, perihelion is not a good name for the point on the orbit that is closest to the Sun, which also makes "peristice" mostly meaningless, even if much better than "solsthelion".
Peri- means "around" (like in "perimeter" or "periphery", where the Latin translation of "periphery" is "circumference", or in "periscope", which means "look around"), which does not suggest much in terms of distance. For instance, both the Earth and Neptune move "around" the Sun, but at vastly different distances.
Apohelion for the most distant point on the orbit is a better name, because apo- means "away", but it still not the best, because "away" refers to direction of movement towards exterior, and not to the distance from the center. A body can be moving away, but still be very close to the center.
The Greek prefixes meaning "close" and "distant" are "anchi-" and "tele-" (first "e" is long). Therefore much more appropriate names would have been "anchihelion" and "telehelion", for the closest and for the most distant points on the orbit. The prefixes "anchi-" and "tele-" have already been used in the "Iliad" and "Odyssey" of Homer, e.g. for "close combat" (like with swords) and "distant combat" (like with arrows).
"Solstice" is a better name, because "-stice" is cognate with "stay" (and with "stand"), and it refers to the fact that during a solstice the apparent movement of the Sun on the sky has a minimum speed, it almost stays in one place. So, for the time interval when the Sun both has a minimum speed of the apparent motion on the sky and is at a minimum distance from us, a term more correct etymologically would have been "anchistice", which is still not optimal, because it combines a Greek prefix with a Latin suffix. Translating "anchi-" into Latin would result in "propistice" (approximately meaning "staying close").
Peri for close and Ap for far is the existing terminology, and has been since Kepler, who in turn created them following a pattern set by Ptolemy in the Almagest (apogee and perigee are simply transcriptions).
So you are somewhere between 400 to 1800 years late to this terminology fight.
Peri- for close and Apo- for far is not the existing terminology, in any other words except the derivatives of "perigee" and "apogee". You do not say apovision and apophone, you say television and telephone.
The fact that "perigee" and "apogee" have been used for the first time with this meaning by Ptolemy, who was a native Greek speaker (but of Koine, not of classical Greek), is not enough to ensure that they have been good word choices and that they have been used properly, conforming to how they were used by the majority of Greek speakers. For any language there are many native speakers that have poor knowledge about their own language.
The words "perigee" and "apogee" have not been coined by Ptolemy, but they were Greek words that have been in use for about a half of millennium before Ptolemy. For example "apogee" has already been used by Aristotle.
At the authors from before Ptolemy, both "perigee" and "apogee" had been used with their correct expected meanings, i.e. the former as an adjective for things that go around the world and the latter as an adjective for things that go away from the ground.
Ptolemy has been lazy and instead of creating a new pair of compound words, well chosen to express in the clearest way their intended meaning (i.e. closest and farthest), he has just repurposed two existing words, giving to them new meanings that were inconsistent with the original meanings of their prefixes.
Moreover, the guilt is not entirely of Ptolemy, because he had not defined "perigee" and "apogee" with the current meaning, because his planetary model did not contain elliptic orbits where there are 2 well defined extreme points. Ptolemy has used "perigee" and "apogee" in a more vague sense, of regions close or distant to Earth, where a region where something goes around the Earth was understood to be close to Earth and a region that can be reached by going away from Earth was understood to be distant from Earth. Only after Kepler, when "perigee" and "apogee" have been redefined as extreme points of an orbit, their original Greek meanings have become completely inconsistent with their modern definitions.
Kepler coined "perihelium" as latin for "point of an orbit closest to the sun" and published it his 1609 book "Astronomia nova aitiologetos", so insisting on 1680 because that's when the spelling with "on" at the end arose is some real pedantic quibbling.
Surely it's more tempting to go for "it should be 415" is more tempting?
Peri- means "around" (like in "perimeter" or "periphery", where the Latin translation of "periphery" is "circumference", or in "periscope", which means "look around"), which does not suggest much in terms of distance. For instance, both the Earth and Neptune move "around" the Sun, but at vastly different distances.
Apohelion for the most distant point on the orbit is a better name, because apo- means "away", but it still not the best, because "away" refers to direction of movement towards exterior, and not to the distance from the center. A body can be moving away, but still be very close to the center.
The Greek prefixes meaning "close" and "distant" are "anchi-" and "tele-" (first "e" is long). Therefore much more appropriate names would have been "anchihelion" and "telehelion", for the closest and for the most distant points on the orbit. The prefixes "anchi-" and "tele-" have already been used in the "Iliad" and "Odyssey" of Homer, e.g. for "close combat" (like with swords) and "distant combat" (like with arrows).
"Solstice" is a better name, because "-stice" is cognate with "stay" (and with "stand"), and it refers to the fact that during a solstice the apparent movement of the Sun on the sky has a minimum speed, it almost stays in one place. So, for the time interval when the Sun both has a minimum speed of the apparent motion on the sky and is at a minimum distance from us, a term more correct etymologically would have been "anchistice", which is still not optimal, because it combines a Greek prefix with a Latin suffix. Translating "anchi-" into Latin would result in "propistice" (approximately meaning "staying close").