London System Theory - c3 vs c4

This position arises often out of the London System games. Here, I prefer to play c3, to avoid black from playing Nb4 after I play Bd3 later. But I often see players play c4. Which is the better move here and why?

c4 is by far the best move at it strikes in the center. Nb4 is not dangerous at all here because the bishop is still on c8. If Black had Bf5 in that would be another story. White can prepare Bd3 with a3 if that's really necessary.

If I play c4, then I imagine it is better to play Nbd2 before Bd3 right?

It is probably right to delay Bd3 for the time being but as c4 is aimed at striking against d5 White should seriously consider putting the N at c3 instead of d2.

Aha okay.

I am wondering if when I play Bd3 at some point and opponent responds by playing Nb4, I should move the Bishop to b1, or let the opponent exchange it..

Bb1 is almost always right. Nc6 without c5 is kind of dubious in the first place and both c4 and c3 should lead to a good position. White just needs to stop e5 in the c3 lines

In general the exchange should be avoided as the Bd3 is a better piece than the badly placed knight. Of course there are exceptions. However where the bishop should go depends on the concrete position. For instance after 7. Nbd2 b6 8. Rc1 Bb7 9. Bd3 Nb4 the retreat to b1 is the only good move, whereas after 7. Nc3 Re8 8. Bd3 Nb4, Bb1 isn't very good because the rook is still at a1 and the pawn at c4 is actually hanging, so 9. Be2 is the best bishop move (and for this reason it may be better to start with 8. a3 or 8. Rc1 rather than 8. Bd3).

Thanks!

Would you prefer c4 > c3 also for the following position (the same with a small change) - 

 Yes because here c3 would give Black the extra option of playing Qd6 followed by the e5 break (which White should stop as Fizzyband said). The value of 7. c3 in this position depends on the variation 7. c3 Qd6 8. Bb5 (forced otherwise e5) Bd7 9. Nbd2 a6 10. Bxc6 Bxc6 11. Ne5 and White's only advantage is the Ne5 so this looks quite fragile.

@GothamChess seems to prefer c3 here. https://youtu.be/49H728S_VjM?t=98

The bishop is already on d3.

Aha. Thanks!

While not my thread, I did learn important things from your post concerning the pawn structure. Thank you! By removing the pieces I was able to see more clearly. Appreciated @IMBacon.

+1 I also much appreciate the advice from  @IMBacon. Your explanation about the pawn structure and associated pawn breaks goes a long way to explaining this mystery about c3 versus c4... thank you!

Here I do prefer c4 because Bb4+ combined with Ne4 and the latent Qa5 does not form a pressure cluster on d2 anymore, and we can safely castle.

In many other lines, c3 is the better move to play in these London positions. It creates a stronger pawn structure and instead of holding the center we can focus on kingside attacks.