1.d4 d5 2.c4 e6 3.Nf3 Nf6 4.e3 a6 
- Thanks for that idea! I had not seen it before.

1.d4 d5 2.c4 e6 3.Nc3 Nf6 4.e3 
- I would not hesitate to play 4...c5 there, but maybe you don't want to play an IQP position. Is there anything wrong with 4...a6 again?

In the Nimzo, black's ...Bf8-b4 fights for the e4-square. Once white has commited to e2-e3, I don't see the need for ...Bb4.

In the Ragozin, against white's e2-e3 and maintaining the c4-d5 tension, black often goes (in some order), ...d5xc4, ...Bb4-d6, ...a7-a6.

Putting those two thoughts together, one plausible idea is 4...a6 5.Nf3 Nc6!? 6.Bd3 (maybe 6.b3 is better) 6...dxc4 7.Bxc4 Bd6 8.O-O O-O 9.e4 e5. Here black is a tempo ahead of a Taimanov Nimzo-Indian (4.e3 Nc6). 5...dxc4!? would be an actual Taimanov Nimzo-Indian -- black saves ...Bf8-b4-d6, white saves Bf1-d3-c4.