an ordinary chessplayer wrote on 09/16/22 at 21:40:52:

re 5.f3: I wasn't aware Black needed an improvement to claim equality. Simply 5...exf3 6.Nxf3 g6 intending Bg7,0-0,Qb6, Bf5 etc gives Black a lively game if White plays 0-0-0 and h4. But I'm no BDG specialist and their evaluation + current theory standing is welcome.
re 1.d4 d5 2.Nc3 c5: Not for 1.d4 d5 players who prefer to equalize without taking the chance of a sharp game. Still, this move can surprise Veresov players, and doesn't look bad at this point.
Back to OP:
re 1. d4 d5 2. Nc3 Nf6 3. Bg5 Nbd7 4. Qd3 c5: if 5.0-0-0 c4! is good for Black (as it looks), does White have to play 5.e4 instead, or 3.Bf4 ? 
re 1. d4 d5 2.Qd3: I'd be surprised if White could even maintain equality after 2...c5

@aoc: sorry, mixed up the early moves, thanks for pointing out.
I meant: 1. d4 d5 2.Nc3 c5 and now:
- if 3.e4 then 3...dxe4 4.d5 Nf6 =
- if 3.Bf4 Nc6 4.e4 etc

I'll try to edit my post if allowed. Post edited to start the main line with 3.Bf4 Nc6

As a Chigorin player, I noticed many years ago that 1.d4 d5 2.Nc3 c5 was playable/equal for Black (hope I'm not wrong!) and always came as a surprise for White. All this surfaced when @FizzySoda spoke of Veresov (ok) and reversed Chigorin (not sure). Yet, I'd be happy if 1.d4 d5 2.Nc3 c5 still holds !?

Reverse Chigorin (or reverse Albin) doesn't sound great, as the Chigorin is a react opening where Black need to double pawns somehow. However, Veresov players must know that after 1.d4 d5 2.Nc3 Black can play the solid 2...Nf6 3.Bg5 Nbd7 but also the funny 2...c5. Play often continues 3.Bf4 (or 3.e4 dxe4 4.d5 Nf6 =) 3...Nc6 4.e4 cxd4 5.exd5 dxc3 6.dxc6 Qxd1+ 7.Rxd1 bxc6 (not 7...cxb2? 8.Bb5 or 8.cxb7 and the game ends in tears and single digits) 8.Bc7. Here Black must be a bit careful (see diagram):
a) 8...Bb7 looks awkward after 9.bxc3 Nf6 10.Rb1 Bc8 but the position can become double-edged if Black is allowed to play Nd5 and Bg7
b) 8...Bg4 9.Nf3 looks OK (by computer) as long as Black plays 9...a5! to protect against Ba6-b7 
c) 8...e6 looks simplest, as Black can counter 9.Re8+ Ke7 10.bxc3 with 10...Nf6 11.Nf3 Nd5 12.Ba5 Bb7, and if White is more ambitious with 10.b3, Black can play 10...Nf6 and e.g. 11.Ne2 Nd5 12.Ba5 c5 13.g3 Bb7 with the plan of playing g6/Bg7/Rc8 and ...c4 while White regains the c3 pawn.

In fact the best bet for Veresov players may be the Prie/Jobava setups with d4 Nc3 and Bf4 instead of Bg5. I hope the move order with 1.d4 d5 2.Nc3 c5 3.Bf4 Nc6 is not suboptimal for Black regarding current Prie/Jobava theory, which I don't know?

EDITED to correct move order issues pointed out by @aoc

Or maybe 5.e4
I remember 5...cxd4 (the other move is 5...dxe4) 6.Bxf6 Nxf6 7.Qxd4 dxe4 8.Bb5+ Bd7 9.Bxd7+ Qxd7 10.Qxd7+ Kxd7 11.0-0-0+ Ke8 12.Re1 as leading to equality