All of them are pretty bad, but I'm going to defend the cold water myth a bit. Fish are cold-blooded creatures, so their metabolisms are dictated almost entirely by the temperature of their surroundings. When reading about aquarium fish in general, I've read that increased temperatures increase the metabolisms and ultimately shorten the lifespans of fish simply because they age quicker. A fish kept at cooler temperatures should require less food and could live longer.
However, all of this is ignoring other variables present. For example, a tropical fish kept at temperatures cooler than it requires will have a weakened immune system and be more susceptible to disease. So, odds are that if you keep your fish at lower temps than it needs it could die young due to any number of diseases. Not to mention that it also seems unnatural to artificially prolong the lifespan of a fish beyond what its body is designed to provide for.
I think the cool temp betta myth must have come from people who show bettas. I've read that many people keep their best show bettas at cooler temperatures to prolong their lives so that they can get more wins for the fish. Still unnatural if you ask me, though.