Comparing a computer generated image to woven cloth is always going to be problematic.

The best way to get the specific colors you want is to be as specific as possible when ordering from the mill. Sending them a sample to match is the best option, obviously.

I note that in the STA's notes on this tartan, Peter MacDonald is listed as the source of the information. As Peter frequents this forum, perhaps he will be able to comment on the colors.