Let me see if I can clarify at least some of the classifications for bettas....

Cambodians, if I remember right, is a certain color pattern - white or light body, with deeply colored finnage.
The double tail, male and female, is a betta with shorter tail, not like the two in your signature, and the tail is split in half. You can see it very good on the picture you found with the green betta.
Crowntails have long fins as well, with the tail looking like the indentations of a crown, like here in this link....
crowntail betta Its very good to see on the picture, the forks that stand up from the tail....
Then we have a few more, like Mustard gas, which are the opposite from the Cambodian - colorful body, beige or light coloration in the finnage.
I hope that helps you a little bit!!