Bollywood is not synonymous with Indian. "Bollywood" is one of many entertainment industries in India. "Indian" is a nationality. Some Indian actors work in Bollywood; many do not. Some actors who work in Bollywood are Bollywood stars but some are better described as actors who occasionally work in Bollywood film. Most actors who work in Bollywood are Indian but not all of them are.
Here's a handy cheat sheet:
* Kunal Nayyar from the Big Bang Theory and Freida Pinto from Slumdog Millionaire are Indian actors but do not and have not worked in Bollywood. (Not Bollywood stars.)
* Kal Penn of Harold and Kumar, Aziz Ansari, and Mindy Kaling are all Americans. Archie Panjabi aka "Kalinda" and Dev Patel from Slumdog Millionaire are British. None of the above have worked in Bollywood film. (None are Bollywood stars.)
* Kumail Nanjiani is Pakistani. (Not a Bollywood star.)
* Irrfan Khan is an Indian actor who sometimes works in Bollywood but is better known for his artsier films. (Not a Bollywood star.)
* Anupam Kher is an Indian actor works in Bollywood ALL THE FREAKING TIME but he's better known a character actor playing fathers, crazy uncles, and villains, etc. I suppose you could call him a "Bollywood actor".
* Purab Kohli from Sense8 has worked in Bollywood but it's a huge stretch to say "star." He's much better known as a former MTV VJ. (Yes, they still have those in India.)
* Pitobash from Million Dollar Arm is an Indian actor who, like Irrfan Khan, is known for work in artier films. (Not a Bollywood star but he is a sweetheart.)
* Anil Kapoor and Priyanka Chopra have starred in many, many Bollywood films are legitimate Bollywood stars.
Dhanush is Indian and has starred in one legit Bollywood film (Raanjhanaa) and acted in a great Hindi language film (Shamitabh) but he's not a Bollywood star. His main fanbase and most of his best work is done in Tamil films, not Bollywood. You can call him a Tamil film star if you must but for a middlebrow arthouse film like The Extraordinary Journey of the Fakir is set to be, why not respect him as an actor and just say "award-winning Indian actor Dhanush" or something and perhaps list some of his films. He has some great ones; his performance in Maryan was phenomenal for a start.
When Bollywood is used as a synonym for "Indian" it's not only inaccurate and lazy reporting but it also erases the humanity of actors of South Asian descent working in Hollywood and turns them into caricatures. Years of lazy and biased criticism have turned "Bollywood" into a huge punchline in the West and, sadly, that's not going to change anytime soon. I've stopped expecting useful discussion of any popular Indian film out of the Western media but at the very least they should be treating the actors who make those films (and the people who happen to share an ethnicity with the people who make those films) with some respect and dignity. Throwing the word "Bollywood" around at anybody with brown skin is not a useful descriptor.
The point is this: Bollywood is one of many film industries in India; Bollywood is not an ethnicity.