If the tank can hold 2gallons of water or more then a betta would be fine in it. Betta's are not very active fish, they can live with less space than other fish around their size like guppys need- a tank/bowl/container of 2gallons or more is fine for a betta.
Filtration is not nesarsary as long as daily water changes are performed on the tank to ensure the water stays fresh and ammonia levels do not rise. Betta's have a primitive pair of air breathing lungs which allows them to take oxygen straight from the air at the surface of the water, meaning that DO levels in the water are not too important. The main thing when doing daily water changes is to make absolute certain that the new water has the same temp and ph as the old water in the tank, and that if you do a 100% water change that the fish is carefully acclimatise back into the tank after the water change is done.
Heating is generally advise though, since although betta's do fine in temps around 23-29degree's, however like most fish daily fluctuations in temp are stressful for them and most people do not have rooms in their accommodation whose temperature does not fluctuate daily from day to night etc, so heating is good as it helps prevent fluctuations in tank temperature.
About the betta's and puddles thing: This is true and not true at the same time. In the betta's natural habitat, it experiences dry and wet seasons- during the wet season, the betta may have a whole giant pond to itself, but during the dry season it may have just a puddle- due to this betta's have evolved to be very tough and adaptable fish, it doesn't really matter if you give your betta a 2gallon tank or a 22gallon tank, the main thing is that the water quality and tank is kept clean, the tank has a very gentle or no current in it (betta's don't cope well with medium to strong current/water flow in their tanks since the long fancy finnage so many betta's have been bred to have weights them down and tires them out etc), the temp is kept stable and warm and the tank has some decor in it (preferably planting, whether real or fake, since betta's dwell in densely planted area's in their natural habitat and prefer being in planted area's when given the choice etc).
A single betta is the only fish i would advise for a tank as small as 2gallons- don't put any guppys, tetra's, platys etc in there. Apple snails would do good in the tank as although they come from a different part of the world as to betta's, their natural habitats are quite similar. Shrimp would not be a good idea unless you have filtration in the tank as shrimp are very sensitive to water quality and would not cope as well as apple snails or betta's would living in an unfiltered tank.