It's all down to who is the actual operator rather than the authority.
Green, Pink and Yellow are operated by BTSC (as BTS, NBM and EBM) - Rabbit
Blue and Purple are operated by BEM - MRTplus
Red is operated by SRT - Their own card, can't remember the name they use.
I work for one of the operators (evidently not SRT).
It can get horribly confusing for the public, particularly when many refer to any elevated railway as the BTS or Skytrain. Blue, Purple and Red are all, at least partially, elevated but only Green is actually BTS.