In the original theatrical release the color correction within the Matrix was intentionally a sort of "bleach bypass" look - not as dramatic as what you see amateur photographers doing but definitely quite different than the heavy green hue. Personally preferred this look myself and I wish they had kept it for the subsequent films. If the Matrix is supposed to look like "our" world then why would it be heavily green ? It still contrasted very well with the cold, metallic muted colors of the "real". The greens were mostly only in the shadows and very subtle and in some scenes more subtle than others.

The reason the original looked so good and is hard to replicate now in post is that they were very likely applying actual chemical color corrections to the original negative film. Probably using some kind of transfer film and a variation of the bleach bypass or similar method.