I really don't see what looks like rust in those pictures. The grout may be stained with something else, but I don't think that's rust. I would apply a good tile and grout cleaner, scrub the grout lines with a grout brush and clean as normal. I suspect that will come out. A floor pad won't really get into the grout lines very well, but will scrub the tiles really well for cleaning. You need some type of brush for tile and grout to get into the grout lines effectively if you use a floor machine. Those are sold by a number of vendors just do a search for them. If you decide to go that route you might also want to buy a splash guard for your brush to prevent splashing on to walls or appliances. Your customers won't like that and tile cleaners can do PERMANENT DAMAGE to some appliances and furnishings really quick. If the job's not too big a simple grout brush you can buy locally will do the trick. I hear the one sold at Home Depot is pretty decent. I believe the Shark tile and grout brush may be the best on the market and it's fairly cheap as well, but you will need to get it online somewhere. It is sold at numbers of outlets including Amazon and many others. I personally have used a product sold locally at Lowes to remove rust stains in particular from ceramic tile. It is NOT for natural stone, period. It is ZEP Tile and grout cleaner and brightener. I keep a quart on hand for use on man made tiles only. It can do permanent damage to natural stones.