The original DVD was way, way less green than later releases, which were changed to match the more-extreme greens used in the sequels. IDK if it was as subtle as in the theater (I did see it there, but most of my watches were the first-run DVD) but it was far subtler than later DVD printings, and all but IIRC one fairly recent blu-ray that finally dialed it back to something less eye-searing and at least close-ish to the original.
The original has a green tint to the matrix scenes, it's just relatively subtle and blends into a general coolness to the color temp. The heightened green from later home printings is really, in-your-face green, to the point you don't really notice the coolness, just greeeeen.
The Matrix is an interesting one because it really caught on with the DVD release. So that was most peoples first exposure to it, not the theatrical release. Even if incorrect, if that was the first way you saw it, it is likely how you consider it "should" look.
It's a bit disingenuous to imply The Matrix did not catch on until DVD release. The Matrix broke several (minor) box office records, was critically hailed, and an awards darling for the below the line technical awards.
Having said all that. One of the most interesting aspects of conversations around the true version of films and such is that just because of the way time works the vast majority of people's first experience with any film will definitely NOT be a in a theater.
I didn't meant to say no-one saw it theatrically but I probably did undersell it there.
The DVD was such a huge seller and coincided with the format really catching on. The Matrix was the "must have" DVD to show off the format and for many was likely one of the first DVDs they ever purchased.
It was also the go-to movie to show off DivX rips.
The popularity of The Matrix is closely linked with a surge in DVD popularity. IIRC DVD player prices became more affordable right around 2000 which opened it up to more people.
Noodle made a charming video about going mad researching this: https://www.youtube.com/watch?v=lPU-kXEhSgk