With 4.d3 White quietly defends the e4-pawn from attack, and can either continue in "Giuoco Pianissimo" style, aiming for quiet development, or can continue by analogy with the closed lines of the Ruy Lopez, aiming to re-route the c4-bishop back to c2 and aim for c2-c3 and an eventual d3-d4.
My Gambiteers' Guild site coverage is here: http://tws27.weebly.com/two-knights-defence-4d3-4nc3.html
I feel that it is important to cover these lines, since many sources on the Two Knights Defence focus on White's aggressive tries with 4.Ng5 and 4.d4, which tend to allow Black easy counterplay, and so those who are attracted to Black's counterattacking possibilities may get depressed when White plays 4.d3 and aims for a closed manoeuvring game. I don't advocate 4.d3 as it is rather against the spirit of my chess site (Dr. Dave Regis at Exeter Chess Club calls this sort of approach the "Old Stodge", especially as most club players don't know how to play Closed Ruy Lopez type positions in an ambitious way). Thus, rather than the "encyclopaedic" type approach that I used in the case of 4.Ng5 and 4.d4, I have mainly focused on ways to generate aggressive play from Black's point of view.
The most obvious way to generate active play is 4...d5 but unfortunately this is somewhat dubious, since after 5.exd5 Nxd5 6.0-0 Black has no good way to defend the pawn on e5 (approaches with ...f7-f6 tend to be both passive and weakening). The gambit line with 6...Bc5 7.Re1 0-0 8.Nxe5 Qh4 might work at rapid time limits but White can get a large advantage with 9.Qf3, or even 9.Nf3!?, inviting Black to take on f2.
Therefore Black does best to refine the plan by castling short quickly (to get the king off the e-file) and only then playing ...d7-d5. 4...Bc5 is the most active-looking post for the king's bishop, taking aim at f2 and envisaging 5.0-0 0-0 6.c3 d5, but 6...d5 doesn't work well against 6.Nbd2, while Black also has to have something ready against 5.Nc3, preventing ...d5. The Canal Variation (5...d6 6.Bg5) is best met by 6...Na5, rather than 6...h6 7.Bxf6 Qxf6 8.Nd5 Qd8 9.c3, which gives White chances of an edge because of the plan with d3-d4.
It seems that Black gets more scope for active play with the apparently-restrained 4...Be7, recommended by John Emms in Play the Open Games as Black. In most lines, approaches with an early ...d7-d5 are playable (the line 5.0-0 0-0 6.c3 d5 is particularly effective here) and sometimes Black can offer a Ruy Lopez, Marshall Gambit-style sacrifice of the e5-pawn in return for kingside attacking chances. The ...d7-d5 plan doesn't work against 6.Re1, but in that line Black can consider ...Kh8, ...Ng8 and ...f7-f5, striving to open the f-file and take advantage of the fact that Rf1-e1 weakens the support of the f2-pawn. The 5.Nc3 variation can be met by ...d6 and ...Nf6-d7 envisaging ...f7-f5.
An alternative idea for Black is to leave the king in the centre for a while and push the kingside pawns towards White's king, which I have experimented with in my own games. The bottom line is, it works well if White plays passively, but is dubious against White's more ambitious approaches, and queenside castling can run into a dangerous attack with a2-a4 and b2-b4-b5. I have thus restricted myself to referring to a few high-level practical examples of this approach in the notes.
My next update will focus on the Boden-Kieseritzky Gambit (which will most likely conclude my investigations into the Two Knights Defence), which is likely to feature a return to the "encyclopaedic" style of coverage, since some readers might also fancy trying it out from the white side. After that I will most likely investigate the Evans Gambit with 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.b4.