In regards to the game played, I think Black can improve in this line by avoiding 8...Bxf3.
I disagree that White has a considerable edge after 4...g6 f4. Similarly I don't think that White has much better after 4...c6 other than to transpose into the Flohr Variation of the Modern with 5.Nf3.
Perhaps this could be named the Astengo Variation, since he (Corrado Astengo(2134) of Italy) appears to hold it in some regard. Some of my notes on the most recent games in this variation (I've checked and rechecked for typos as I wasn't able to cut & paste and keep my own formatting -- so hopefully there are no big errors)......
1.e4 Nf6 2.e5 Nd5 3.d4 d6 4.Be2
4 5 6 7 8 9 10
1 ... dxe5 c3 Nf3 Qa4(a) Bb5 Qf4
dxe5 Bf5 Nc6 e6 Qd7 Nb6 a6 =, #1
2 ... ... Nf3 0-0 c3 Qa4 Bb5
... Nc6 Bg4 e6 Be7 Bh5 0-0(b) unclear, #2
3 ... ... ... ... Na3 c3 Rxd1
... ... ... Ndb4 e6(c) Qxd1 0-0 +/=, #3
4 ... f4 Nf3 c4 h3 exd6 Nc3
g6(d) Bg7 0-0 Nb6 f6 exd6 Re8 =, #4
5 ... c4 exd6 Nc3(e) Be3(f) Nf3 0-0
... Nb6 cxd6 Bg7 0-0 Nc6 Bg4 =, #5
6 ... Nf3(g)
c6 Bg4(h) unclear, #6
a Black is also doing OK after 8.0-0 a6.
b Unclear, although this game continued 11.Bxc6 Bxf3 12.gxf3 bxc6 13.Nd2 Qe8 14.c4 Nf4 15.Ne4 Ne2+ 16.Kh1 f5 17.Ng3, =.
c The immediate exchange of Queens appears to offer more for Black, 8...Qxd1 9.Bxd1 (9.Rxd1 Nxe5 10.Nxe5 Bxe2 0-0-0 and White loses a pawn) 0-0-0 =/+.
d Invites 7.Nf3 and transposition to the Alekhine: Modern, Alburt Variation. 5...f4 Bh6 may be interesting to consider.
e 7.Nf3 transposes to the Alekhine: Modern, Karpov Exchange.
f 8. Nf3 transposes to the Alekhine: Modern, Alburt Variation.
g Possibly unsatisfied with the performance of 4.Be2 or prefering book lines vs. 4...c6, Astengo returns to the Alekhine, Modern.
h Alekhine: Modern, Flohr Variation by transposition.
#1
Astengo(2191)-Horn(2272), Open, Geneva SUI, 1999.01.27, 1/2
#2
Briggs(2200)-Crouch(2407), Smith & Williamson ch-GBR, Scarborough ENG, 2001.08.02, 0-1
#3
Dogge-Goverde, HZ Open, Vlissingen NED, 2001.08.02, 1-0
#4
Astengo(2164)-Ermeni(2275), MTO, Biel SUI, 1999.07.23, 0-1
#5
Houska(2251)-Summerscale(2423), 86th ch-British, Scarborough ENG, 1999.08.12, 0-1
#6
Astengo(2106)-Zelcic(2500), 9th Aosta Valley, Saint Vincent ITA, 2001.02.03, 1/2
Nothing earth shaking there, but maybe it's useful.
Alfred