I asked my tiler which he would rather do: remove the old grout and re-grout, or use the colorant. He said re-grouting would be easier for him (remember, he's doing this for free). We saw him do it, and it wasn't as difficult as I had thought. They just took it out with a utility knife. They did about 50% of the floor, since we still had stuff on it that was too hard to move at the time. I bought a $400 Fein oscillating tool for the job, and I plan to remove the rest myself. The other argument is that you could apply the colorant yourself, one section at a time, without paying for more labor. When I do stuff myself, I take forever, but I do it about a million times more carefully than any worker seems to care to do. Grout seems to be the black hole of flooring knowledge. Daltile is a very good high-line supplier, but another tile supplier was able to solve most of our problems (mainly, finding the Hydroment grout samples there). At first we had a lot of effluorescence, but they had the product that removes it effectively there as well. Good luck!

I would take a different route. For small shifts, you can use a glaze that is a heavily pigmented product that you apply between coats of finish. But you can do moderate, not extreme shifts with a glaze. Some people use a gel stain as a glaze. Once the glaze has thoroughly dried, apply a top coat compatible with the glaze and under coat. You can also use a toner, that is a finish with color in it. The real issue with this is it almost has to be sprayed to be even. You also need some decent experience with a spray gun. You can also effect smaller color shifts but a large shift might leave you looking at an opaque layer that looks like paint. I consider the all-in-one products like Polyshades (r) technically a toner, but it is very difficult to brush on without streaking. But your first thing is to find out what kind of finish is on there now. Chances are it is a lacquer of some type if it is a factory finish. Shellac hasn't been used much commercially for sixty years, and varnish is a slow and craftsman-applied finish and would be quite rare for a production piece.

With out showing your actual materials, I'll say stick to the marching color. There are different versions of the grout you can use. Post what you were planning to use.

Its either still wet or the light piece is from another color lot. Hopefully just wet and it will take a few days. Never purchase from big box retailers without checking all pieces for acceptable color matches.