Bettas can live, and if you stumble upon a hardy one for a good couple of years, in a small unheated container....but this is no quality life, and I'm sure though that one you had hung on for a while you had a few others that didn't last quite nearly as long.
A dog can survive in a cold, dirty, small cage for quite some time, but that doesn't mean its a good environment. Its the same thing. Bettas can survive in a small, cold container but they don't ever thrive.
Contrary to the popular myth passed around, betts are in fact a true tropical species of fish, meaning that in order to live a full, happy, thriving life they require a regular. stable temperature of 78-81 degrees Fahrenheit 24/7 year round.
In most places, even if only to keep the water temp STABLE as fluctuations can be stressful, this does require an aquarium heater.
What you want to look for with a heater is something fully submersible, and adjustable with a thermostat. STAY AWAY from present heaters and pad-like heaters that you can't adjust, they can easy under/over-heat and are very unreliable.
Hydor makes some fantastic heaters, just about everyone I know of that has one loves it. Elite is also a great brand, you can find them for a good price on Amazon(they say they're present heaters, but they're actually adjustable). Personally I'm a fan of the(adjustable!)Aqueon brand heaters and Petcos own brand that they sell, both have worked well for me.
Bettas are also naturally very active fish, and they do require much more space to stretch their fins then a little bowl or vase. Along with a heater, yes, you will want to upgrade your tanks(something large is much easier to stably heat anyway). Anything 2.5+ gallons will work, 5+ gallons being ideal of course.
Aqueon is actually having a wonderful sale on their 2.5-5 gallon aquariums.
You'd have to remove/baffle/replace the filter probably, but its a great deal. ;)