Should tile floors be cleaned with vinegar only? Is the advice against using other solutions a myth?
In the to-do section of the care instructions you posted, it suggests "Use an all-purpose, non-oil based household or commercial cleaner."  OR, it says, you can use vinegar/water.  

I think you are over-analyzing.  The normal floor mopping solutions like Pine-Sol or a Swiffer will work fine.  Just don't use anything abrasive that will scratch the tile or an aggressive bathroom "tile/grout cleaner" that is meant to bleach and renew grout that is stained.  

If you're concerned about a particular cleaner, you could always test a small section that is out of sight.  You can also apply a "grout sealer" solution over the grout lines.  This is tedious, but can help slow down staining of the grout.