Just my input here. We do a lot of tile replacements in restoring floors. So we initially started doing this we used to cut the grout with the fein machine and yes it is messy but found it painfully slow no control. Now it is not a problem the Exakt Saw Dc270 is perfect for it at first skeptical beacause it was a JML product at first. but with this saw you can set the depth so if you know the depth of the tile you are going to have no problems what so ever and the best thing when you attach the vac you have virtually zero dust and its quicker. I must admit the blade supplied with the saw is crap but I took a blade of the angle grinder and put that in it but now they have actually bought out a diamond blade.
Grout coloring is good have done a lot of it, but IMO should be as a last resort if possible.
Tony