And it has always been that way though. This is all Rubinstein. ... 0-0, Bd3 ... c5 gives you flexibility to play sharper with Nf3 ... Nc6, 0-0 ... d5, a3 ... Bxc3, bxc3 ... dxc4, Bxc4 and then Qc7-e5 (depending on white as well).

The whole NID become somewhat marginal on higher level since defensive/positional white players can always avoid it by either steering the game into QID or something else but it works wonders against amateurs on all levels. Looks like dominant d4 on higher level is Slav and derivatives or QID among centerless openings.

After all, it was the system pioneered by my favourite player =)

e3 - aka the Rubinstein variation - is meant to avoid doubled pawns by Bd3 and Nge2. Is it necessarily bad after 4. e3, however, to just play 4...Bxc3?>

Yes

Yes>

There doesn't seem much difference to me, really ...

Let's consider the most salient point of the position here, namely White's Queenside Pawns, specifically the triad at c3/c4/d4 (this theme is pretty much the same in most of the Nimzo lines where Black does not play ... d5 and where White must recapture at c3 with the Pawn).

The most significant of these Pawns is the one at c4, which after Black's ... c5 (a move Black should make ASAP - allowing White c5 causes the Pawn structure advantage Black has to virtually disappear) is fixed, and extremely vulnerable - not merely "weak" but rather "sick" might perhaps be the best term here. Black should follow up with b6, Ba6, and Na5 (and also, generally, try as much as possible to encourage White to fix the center with d5), with very strong pressure - White is forced to tie two pieces to the defense of Pc4, and moreover is very restricted on what pieces these might be (the Rooks are unable to do the duty because of the obstructive Pc3, and thus only the Queen, Knight, and light Bishop are available). By contrast, only one Black piece is really put out of play by this, namely the Knight at a5; the Bishop can easily return to the fray via c8, and as in all cases with a weak Pawn, has the initiative over White's defending piece in doing so. However, Black's Knight does serve another useful purpose at a5; by being extremely difficult to drive off - the only White piece being able to do so being the Knight, and Nb3 is only possible if both Queen and Bishop cover Pc4 - it sets to naught any White hopes of taking advantage of the b-file by playing a4-a5 to lever against Black Pb6, the latter thus serving as a granite roadblock to the file.

In light of these Pawn structure considerations, it seems that the difference between 4 a3 Bxc3+ and 4 e3 Bxc3+ is merely a tempo for White. Specifically, we see at once that White cannot afford Black to be able to place four attackers against Pc4 - if he can do this, the Pawn is lost. Additionally, Black does have these attackers to spare, for he has two Knights (one of the cases in which two Knights are actually better than the two Bishops, simply because two Knights can both be focused on a weak point, while two Bishops cannot both defend it) and, once White's Queen develops (unless to c2), the Queen can move to a4 via d7.

Therefore, in many cases, at some point White will want to play a4, in order to forestall this intrusion by the Black Queen. As such, in the 4 a3 variation, White has frequently wasted a tempo with a3, while he has not done so with 4 e3.

But to me, it seems that a single tempo, especially one whose loss will not be felt for several more moves (perhaps never - if White plays against h7, he might play Bd3 and Qc2, where the Queen still prevents ... Qa4) is hardly enough to consider one variation significantly superior to another. This is especially true considering that White may also wish to play e4 at some future point (which means that then 4 e3 becomes a wasted tempo itself!).