In this position via the Classical move order 6.d3 is just played about 40 times in Mega 2012, while 6.d4 is played more than 1500 times. So the main portion of games you get is indeed coming from the Berlin 4.d3 move order.

Thanks Chessexplained,

That makes a lot of sense since the main move after 4.d3 is Bc5. So I guess that in the move order I suggested, 6. d4 is the best??? Transpositions can be tricky, I often get caught out playing correspondence chess looking up databases.

I would like to find out about the idea behind overprotecting e4 with d3 in the Cordel defence. I hope it will spark some interesting discussion.

After 4.c3 Nf6 5. O-O O-O, the Chess Tempo database for all players shows that White plays 6. d4 four times more than 6.d3. However, 6. d3 becomes more popular as ratings increase; the ratio becomes about even where all players are above 2500. Then the database shows players with ratings 2700 and above are playing 6.d3 three times more than 6.d4. 

I can't compare this with other databases because Chess Tempo is the only one that I know groups ratings by 100 points after 2200. However, 365Chess.com does show a marked increase in the popularity of 6. d3 in its Masters database compared with its Big database.

I can only deduce that top GMs think that 6. d3 plays out better. What is the idea behind overprotecting e4 when there does not seem to be any threat? With 6.d4 the Knight on f6 can't take e4 because the Bishop on c5 is attacked. After the Bishop moves away White can just play Re1 or Bg5.