It may be that something is still stuck in the tile causing that discolouration, possibly old wax, sealer or oil.
Either use that Lithofin cleaner again with a steamer or you could try Lithofin Wax off which does a better job at removing things like that than the detergent.
I would also use an acidic cleaner on the tiles after you have tried the two mentioned above. Acidic cleaners remove things that the strippers mentioned above don’t but the detergents need to be rinsed off or the acid cleaner won’t work.
Agitate with a little scrubbing brush, give things time to soak and be sure to thoroughly rinse all chemical from the tiles. So don’t let things dry dirty.