In what way would this be considered a שינוי? it is the normal way to filter it at all times!
I really don't understand why you are still arguing, Did you look in the Shary Tshuva I referred you to? It's on the daf in Hilchas Shaabos 319
where you quoted from. He talks in a very similar situation when someone attaches a cloth over the bottle or barrel when he fills the cup, same goes for faucet. It all depends where you place the cloth, if on the cup you're pouring "into" , it's דרך בורר indeed, if on the bottle you're pouring "from", it's not דרך בורר.
Besides for that, turning a faucet on is in no way a מעשה ברירה, and the fact that you do it everyday certainty doesn't make it so. It is
NOT the normal way of filtering in any way, it's just the normal way of turning a faucet on, and by chance you get filtered water everyday. (no need to use the big thumb for this
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