1.g3 - d5
2. c4 does this belong to the Reti or Benko or... ?
It's a reversed type of (old) benoni.
1.g3 d5 2.c4 is (A00) Benko's Opening
The line shown above is the closest we have found to a 1.g3 Mainline. 1.g3 is basically a transpositional tool. It allows White to chose an opening after seeing what Black is planning. And of course it also allows Black to have at least a little influence in White's decision.
After 1.g3, transpositions abound.
Benko/Hungarian mainline
source:
Chess Opening Essentials - Vol 4 - 2010 - Djuric - Komarov - Pantaleoni
1.g3
Frequently transposes to
English
King's Indian Attack
Reti
Catalan
1...d5
2...
f6
2...g6
2...c5
2...e5 usually transposes to the mainline
2.
g2
2.
f3 prevents 2...e5 and transposes to Reti/KIA.
2...e5
Here we have a Modern Defense reversed with an extra tempo. However, the authors say it is not enough to guarantee White the advantage.
2...c6 is a popular choice and
can lead to a distinct line
which does not transpose.
3.d3
f6 4.
d2 e5 5.e4
d6
6.c3 (6.
f3 is a KIA)
6...O-O 7.
e2
e6
Makropoulou vs Kouvatsou 2004 1-0
3.d3
A) 3.c4 exploits the extra tempo
and leads to complexity
3...c6 4.cxd5 cxd5 5.d4 e4
Markowski vs Krasenkow 2009
B) 3.
f3 takes inspiration from 
the Alekhine Defense
3...e4 4.
d4 c5 5.
b3 c4
6.
d4
c5 7.c3
Grigorian vs Alburt 1977
3...
f6
4.
f3
f6 is a reverse Pirc
4.c4 d4 5.
f3
f6 is a Reti/Advance variation
4.
d2
c6 5.c3 a5
Sooner or later White should play
f3
creating a reverse Pirc/Classical though
the extra tempo is not significant.
5 moves