Tile – Cloudy white film spots on surface of stone sealer

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.

If you are not sure about whether a your paving is capable of withstanding the jet of water from a power washer, test a small hidden area first.

Natural stone and slabs have the highest resistance to power jet water stream. It is not uncommon to see small pieces of stone or slate taken of by the water stream. You should wear glasses because this can ricochet and get lodged in your eyeball ... ouch.

You need to take care with "wet-cast" slabs and paving.

I would recommend using karchers dirt blaster ...

It has a weird cone shape on the end that actually spins the water at high pressure making the cleaning much more effective.

You should then regulate the distance and angle at your own consideration.

I usually pull away when cleaning the concrete joints as the water stream can easily rip them out. Then do a quick sweep on the tile and only concentrate on areas where tough stains are left. There have been times where small pieces of stone were forced out of the slab and nearly blinded me -- Wear safety goggles!

There is also patio cleaner that does not spew water and debris all over the place and is a bit safer to use on sensitive material.

Power-washing is particularly effective at removing mosses and algae that seem to thrive on certain types of paving, most notably 'textured' patio paving and clay brick pavers. We find that the addition of Jeyes Fluid tends to inhibit the return of this type of vegetation for a short time, whilst not affecting the pavement surface. Certain quality sealants are also reputed to inhibit or prevent algal growth, but are an expensive way to eliminate a minor problem.

You would need to find some local advice on sealants that you can use on the particular slabs you have got. They can work well sometimes -- but also can be a waste of time the rest of times.