I don't know from any official naming conventions for the distance between two pegs.

But I know a naming scheme from Andreas Kleinhans who invented it somewhen in the 80ies.

Here it is.

In the following picture you see a single peg in the middle surrounded by six rings named a, b, c, d, e, f.
On each ring you see some numbers representing those places where you may set a peg.
Other positions on the rings are symmetric and positions outside of the rings are not considered here.


The distances according to Andreas Kleinhans are named after
the Greek Mythology.

Ring a is occupied by Apollo.

On ring b you find the rulers.
1 = Hades
2 = Zeus
3 = Poseidon

On ring c you find the messengers.
1 = Pan
2 = Hermes
3 = Prometheus

On ring d you find the heroes.
1 = Hercules
2 = Achilles
3 = Odysseus
4 = Agamemnon
5 = Hector

On ring e you find the children of Zeus.
1 = Ares
2 = Baccus
3 = Athena
4 = Artemis

On ring f you find the poets and philosophers.
1 = Socrates
2 = Platon
3 = Homer
4 = Zenon
5 = ???
6 = Aristoteles
7 = ???