Yes that's right for the waterchanges!
We didn't quite understand the cycling process when we got our first tank either, but this site us helped alot. No petshop ever explains and just trys to sell you products for your tank.
Cycling is the establishment of good bacteria but not necessarily through having a filter. The bacteria grows on the substrate, glass, tank decorations and plants (and filter media if you have one).
When a fish or axolotl is introduced into a new tank, any food/waste/poo left in the tank creates ammonia (ammonia levels rise, which is the start of the nitrogen cycle). Ammonia will rise, peak and then nitrIte will develop as ammonia drops back gradually to 0; nitrIte in turn will rise and peak and nitrAte may develop (Nitrate can be effected by presence of live plants, so may not readily show on tests). NitrIte will then drop back to 0 and your tank should be cycled. I've found the nitrIte part of the cycling tends to be the longest part of it (waiting for it to finally drop). This is why cycling can take between 3-10.5 weeks. There is really no speedy method to cycle, adding tank decos/media, substrate (which is what you're doing) will help.
Your tank sounds like a much better size, and yes that 20litre tank was a bit small, I can imagine what your levels were. Smaller tanks tend to warm up faster in summer and also tend to get dirty faster.
While your tank is cycling over the next few weeks/month or so; just leave your filter (don't wash it at all - the good bacteria will develop on it and if you wash it, you'll wash away the bacteria).
Invest in a turkey baster if you can get them, if not, ask some of the other Aussie owners, like Teesha, where they got theirs, generally they can be bought from a kitchen section of a supermarket. Turkey basters are excellent daily cleaners and save you having to pull out that siphon for that little bit of waste/poo stuck unde the cave or near the corner. We had to cut the tips of our turkey basters (individual ones for each tank), as they were very narrow and the waste kept dropping and making a mess (can be a bit frustrating when it disintegrates and flies back down the baster too!)
Also, careful what the fish lady recommends. Some people do use snails in their tanks, just be careful what they recommend for an axolotl (generally things like feeder fish, what chemicals to use, or medicines - many fish medicines are toxic to axies).
If you're unsure and the fish lady/petshop person advises you to use something. Just check here first before buying, then you'll save yourself and Otto any stress. Some fish enthusiasts/petshop people, I've met, tend to think that axolotls can be treated the same as goldfish/cold water fish in their feeding/treatment etc...
Good luck and keep us posted!