It helps if one understands a bit of the chemistry here. Start with pure water which pretty much contains nothing but H and O (H20). After that it can contain all sorts of dissolved things, some organic and some inorganic. It can also contain Ions. Ions contribute to the conductivity/TDS of water. Both Nitrite and Nitrate in tanks are in the form of ions (so is most of the total ammonia in the form of ammonium aka NH4).
One can filter out sediment and some other dissolved solids, but the removal of ions requires a de-ionizing resin. I doubt any fridge based filter uses any form of de-ionizing resin. Most RO units do not have a de-ionizing component included unless one is specifically looking for one to be included.
Next, most kits test for Total Ammonia. This is the combination of both forms of ammonia NH3 (very toxic) and NH4 (ammonium) which is way way less harmful. The amount of each type of ammonia in a given Total Ammonia reading depends upon the pH and temperature of the water. pH is much more important here. But, the higher the pH and temp., the more of the ammonia will be in the toxic form.
At a pH of 8.8 and a temp of just 75F, a total ammonia reading of .50 ppm, the NH3 component is .1255 ppm. The red line for NH3 is at .05 ppm for most species of fw fish. Basically, the tap water described above should kill fish fairly fast.
I am guessing that the tap water is being treated with chloramine. When this breaks down, it turns into chlorine and ammonia. If one has a dechlor which also detoxifies ammonia, this is not an issue. But it can make testing inaccurate.
One last observation, different countries have different systems. One cannot assume a lot of things when a poster does not include the country information for where they live. Without that information I will usually refrain from offering solutions ot making product recommendations for dealing with problems because i jave no clue whether they might be available or not in an unknown country.
As for setting up a new tank which is being done to get a bigger one, here is what one should do when the smaller tank has been running for a while safely. Move everything that is in the 10 to the 20 and it should be safe. A bigger tank doesn't make more ammonia, adding more fish etc. does that. So if you do not increase the fish load right after the move, the new tank should be safe for your current fish.
Once the new tank has been running for a while without issues, you can begin to increase the fish load gradually. You can move about 1/2 the water in the 10 to the 20 and fill the rest with dechlorinated tap. This is done so the fish will still have a "taste" of home in the new water. It will not move any nitrifying bacteria. These are in the filter, int the substrate and on other hard surfaces, bear in mind that the bacteria are photosensitive and will be loving under stuff and where there is little or no direct light. this is why filters are not clear.
