I was confused about the comment in your post that the grout dust is the same color as the tile. Is the grout the same color as the tile, or are you suggesting that (perhaps) you're grinding out tile instead of grout?
I'm not the most authoritative source (I've done tiling/grouting but never grout replacement)--but I'd guess that you'd want to remove as much of the old grout as possible. When you grout, you typically want to force in as much grout as will fit between tiles.
Not sure this was helpful or not...but here's to a great looking end-result!