Bile is bright neon green or yellow-green. It indicates small bowel obstruction, because it is coming up backwards from the gall bladder back through the stomach, and that should not happen. It indicates a serious problem, but it does not sound like that is what you are dealing with.
This is a good article on cat vomiting from a vet hospital. There is a bulleted list of serious symptoms if you scroll down:
https://vcahospitals.com/know-your-pet/vomiting-in-cats
This is a more easy-to-read discussion:
https://vetmed.tamu.edu/news/pet-talk/when-to-be-concerned-about-feline-vomiting/