PETA is an extremist organization of radicalist animal rights "activists" whose idea of an ideal world is one in which domesticated animals cease to exist. Their founder is known for condoing activities such as "liberating" domesticated animals, which are incapable of defending themselves, and supporting BSL and feral cat killing projects. What's more, their shelter has a kill rate of at least 80% at any given time, worse than most inner city animal control centers. They keep enough high-profile rescues around to slap fuzzy faces on calendars, and euthanize the rest because they think animals are better off dead than human owned. Their goal is total animal liberation - no pets, no livestock, nothing. And quite frankly, they have a friggen vendetta against the human race. They use lies and misleading information to promote their viewpoints, and trick caring animal lovers into supporting their repulsive organization by thinly disgusing themselves as people who care about animals. And no, I'm not an ultra-conservative meat-eating hunter saying this; these are the words of a vegan animal welfare advocate who happens to have a betta sanctuary.
Now, as for the information, it is true that some species of bettas live in streams and ponds. B. splendens is more often found in rice paddies and marshy/swampy habitats, which - though shallow and stagnant - are typically quite expansive, often at least the size of a foodball field or larger. During the dry season, these bodies of water do become very shallow and in some spots bettas are forced to live temporarily in puddles, which is why they've evolved their fantastic air-breathing aparatus to survive in stagnant water. It is not, however, the ideal or natural water condition for the species, and you can bet your bippy that they are healthier and happier when water is more abundant.
As others have said, however, domesticated bettas can not always handle much space because of their lofty finnage, and often do better in 2.5-5g. I personally am against keeping bettas in less than 2.5g, and prefer to see 5-10g. The minimum for the species currently stands at 1g, though I'd like to see it at 2.5-3g for the sake of the inch per gallon rule, as water quality goes downhill fast in smaller containers, which are also almost impossible to safely heat. So while bettas definately don't need a pond like PETA says, they definately do deserve more space than they are currently alloted by the pet trade.
I understand the point of the campaign - boycott bettas, since most stores and uneducated owners keep them cruelly - but as usual, it is tainted with misinformation. A better campaign would explain proper betta husbandry, discourage buying from stores who do not properly care for bettas, and suggest buying from reputable breeders instead of supporting the mass breeding and "throw-away pet" mentality pushed by pet stores marketing bettas as decorations. Another good idea ruined by PETA's utter impotence when it comes to campaigning, I'm afriad.