Not only it is stored separately, it is also supplied to the International Space Station (ISS) as two different waters. But if water from both sources was consumed, silver iodide precipitate wouldn't form, since the U.S. segment water is deiodinated before it's used for human consumption.
Silver iodide is actually a cloud seeding agent and while I personally doubt it has any serious consequences to our health, the scientific discussion on that front is still ongoing. But, luckily for us, we don't even have to consider that as a possibility, or what happens when you'd drink both Russian and U.S. segment waters onboard the ISS and the silver-iodide precipitate formed. Here's a nice excerpt from an article in Space Safety Magazine:
Since 2008 the SMAT (Metropolitan Water Company Turin) is supplying
the space station with drinking water, truly two different kind of
water for the US and Russian segment. Both segments require a very
stable water, able to maintain its properties for months in zero
gravity, but while the Americans prefer little mineralized water,
treated with iodine salts, the Russians prefer heavier waters, with
more minerals and treated with silver salts. For the Americans, the
SMAT selected the very light water from Pian della Mussa, coming down
from the valleys of Lanzo. For the Russians, the water comes from
wells located in the region of Collegno because of the mineral
content. A sampling of these waters was sent to NASA, which certified
it after thorough checks.
More is explained in the linked to article, but as you see, astronauts and cosmonauts drink different water onboard the ISS, each from their own stored supplies. How strictly this no water mixing rule is enforced, I wouldn't know, but as mentioned, I wouldn't think it would make much of a problem in quantities a person consumes. Or a few of them. Besides, U.S. potable water is deiodinated before consumption, as the article later explains, so that's even less of an issue then, if it's unlikely to happen.
But you can't mix the two separately stored water supplies, complete with their different disinfecting agents, because water is used also for other purposes (coolant, source of oxygen via electrolysis,...) and the silver-iodide precipitate would find its way directly into systems that aren't designed to handle it well. And I believe that's what the article that you're quoting is suggesting.