No, this is a really common myth. And yes those betta are very unhappy. Since betta breathe air from the surface instead of with gills like most fish, people have thought of the horrible idea that they can live in horrible little cubes as decor. Betta come from Thailand, and they NEED a heater that heats to around 78F to be happy, which you can't do in a cup.
If you want a betta, invest in a heater and at least a 2 gallon tank from the start.
A cold betta sits at the bottom and keeps it's fins closed, not moving.
A happy warm betta will be VERY active, and swim around showing off his fins, which is much more attractive than a cold betta in a "designer betta tank". A happy healthy betta will constantly entertain you with a very show off-y personality and they are really a joy to have :)