I'm not sure I've ever seen rust stains on tile I was trying to clean. If your grout cleaning product and process didn't seem to work, I would try a good rust remover. I use the rust stain treatment that Vacaway makes, and have found it to work well on carpet fibers.
I have also heard people report success in removing rust stains from grout using the following products:
---hydrogen peroxide (the strongest are found at beauty
---oxy-bleach, in paste form. Allow 30 minutes to work.
---a paste of hydrogen peroxide & cream of tarter
---a paste of baking soda & lemon juice.
Good luck. Please report back on what worked for you.