The reason for a shop saying 'fine' when the readings aren't is because they don't care if the number for ammonia and nitrite is above zero - if there is ammonia and/or nitrite in the water and the fish gets sick, you'll spend money on medication; if the fish dies you'll spend more money on new fish.
Cycling is the term used for growing 2 colonies of bacteria that 'eat' the ammonia excreted by the fish. These bacteria are in our tap water in tiny amounts - the water company adds chlorine or chloramine to kill bacteria but a very small number do survive. These then have to multiply in the tank till there are enough of them to 'eat' all the ammonia made by the fish. The bacteria are very slow growing and it takes weeks to grow enough of them.
Since you didn't know what cycling means, you can't have done a fishless cycle (where we add ammonia from a bottle to feed the bacteria and get them to multiply before we get fish) so your betta would have gone though a fish-in cycle. During a fish-in cycle ammonia build up so we have to do lots of water changes to stop it harming the fish. After a few weeks, the first bacteria start 'eating' the ammonia and they turn it into nitrite which is also toxic. That too builds up until the second colony of bacteria has grown enough to 'eat' all the nitrite made by the first bacteria, and again we need to do water changes to keep nitrite low.
When the person told you to do complete water changes every three weeks, did you do partial water changes between the complete ones? If you didn't your betta would have been harmed by the ammonia and nitrite in his tank water. And this can cause health problems for the rest of the fish's life.
The tank will now be cycled since it has been running a few months - the snail is fine and they are more sensitive to ammonia & nitrite than fish (or at least the snails we pay money for are; pest snails seem immune to everything)
The best treatment for finrot is water changes. I would do one every day for now. And stop adding aquarium salt. Bettas are fresh water fish and should not have salt in their water. The daily water changes will dilute he salt you have added till it is all gone. Make sure that you add dechlorinator to the new water and warm it to the same temperature as the tank water before putting it in the tank.
The wood in the tank - does it have any rough edges or sharp points? Bettas' fins are delicate and can snag on rough things.
Does any of the decor have tiny spaces? Bettas will squeeze through the tiniest holes and can scrape their bodies - or heads if they try to get through and find they don't fit.