The Battle Chart has not been updated to include the newer dragons so a good rule of thumb is to pick dragons that are good against the magic of the enemy. This is why I put the list on the bottom.
Example: If I'm fighting a Dark Omen dragon, the spooky element uses black smoke magic so using Virtue, Black Swan, Kitsune, Air or Neo Air etc. would be a bad idea since black beats pink and yellow. The better dragons to use would be green and white types since they're good against black.
As far as why the dragons use the magic they use, the game developers determine that and we, in the forums, go with what they chose. Soccer may split into blue/green but they chose the magic to be green vines not us.
In short, treat the bubble magic dragons as blue, vine magic as green, black smoke as black, pink swords as pink, red fire as red and yellow tornado as yellow until the chart is updated one day. If it's not understandable then use the tools not mentioned on the forums.