Here is a crash course on who is a Bengali and who is not (This crash course is very important for non-bengalis who spell kolkata as "kalkatta" and rosogolla as "rusgulla". It's a pain to explain this but here it goes):
Any person whose mother-tongue is any variant of Bengali/Bangla language is a Bengali.
We distinguish the Bengali speaking people of Bangladesh by their nationality/citizenship and call them "Bangladeshi".
All bengali speaking Indian citizens are Bengali Indians
A bengali can be a hindu or muslim or christian or sikh or buddhist or anything else.
A bengali indian thus can be a hindu or christian or sikh or buddhist or anything else
A bengali Bangladeshi can be a hindu or muslim or anything else (if such athing still exists in Bangladesh)
A person whose mother-tongue (or father-tongue) is NOT Bengali is a NON-Bengali.
On a serious note, school text books of different states in India should devote more pages to explain the culture/history of other states in India. Otherwise there will be confusions galore. It is very common to see comments such as "All bengalis should be sent to Bangladesh etc" on Internet. The funny thing is that even in a job interview, an HR (probably a Keralite and a confirmed Christian) once mentioned "Bangladesh" once he knew that the candidate was a Bengali ( + Indian + Hindu). That is height of irony and that is a height of the general lack of education that people of different states of India seem to have received!!
On a lighter note, Bengali gene and culture have infiltrated whole India. Most non-bengali Indians just do not know it
Abhishek Bachchan, Hrithik Roshan, SaifAli Khan are all Bengalis. Most people do not know that. May be they themselves do not know that! It is too late for Indians to spend much time on the subject on "Bengali". "Indian Politicians" cared two-hoots when Bengal was partitioned. And we Bengalis know it well, how the subsequent central govts of India cared for "waste" bengal. I have met callous Indians who asked me in real life "If West Bengal is on the east side of India, why is it called West Bengal?". Those type of guys are either uneducated or I do not know what to say. (And despite all this, there is 0 separatist movement among Bengali Indians) Now it is too late. Just deal with it- Indian President is Bengalee president and Chief of IAF is Bengalee IAF chief. He He