The language spoken in Bangladesh (and parts of North-Western India) is called:
In fact, people use numerous names for both the nationality of people from Bangladesh but also the name of the language. So you might ask: What do each one mean and are they all correct?
Bengali and Bangla can both be an adjective meaning "originating from Bengal" and a noun referring to the Bengali language. Bengali is of Hindi origin whereas Bangla is more purely Bangla. Bangladeshi and Bengalese are only adjectives meaning "originating from Bengal" and do not refer to the language. While "Bangladeshi" is of Hindi or Bangla origin, "Bengalese" is a mix of Bangla and English with Latin roots.
The word "Bengali" can refer to both a person who comes from "Bengal" which is the region of North-Western India and Bangladesh, or the actual language spoken in Bangladesh.
The English language got the term from Hindi "Baṅgālī" (बंगाली) which in turn borrowed the word from the actual Bengali language where it's called "Bānlā" or "Baṅgāla" (বাঙ্গালা).
The term goes much further back, though to Prakit "Baṁga" (বঙ্গ) to the Sanskrit "Vaṅga"(वङ्ग) and probably back to Dravidian roots.
The first time the word "Bengali" was used in English was in a publication from 1762 and it referred to the nationality, and not necessarily the language.
Is pretty much synonymous to Bengali, except for the fact that the English term seems to come directly from the Bengali language "Bānlā" without passing by the Hindi language.
"Bangladesh" is actually derived from the Hindi name for the Bengali country. "Bengaal desh" (बेंगाल देश) in Hindi actually just means "Bengal country".
It is also possible that the term comes from the same thing in the Bengali language "Bēṅgala dēśa" (বেঙ্গল দেশ) but the "desh" part seems distinctly Hindi.
The "i" in the end which turns the word into an adjective could stem from either language.
Bangladeshi is rarely used for speaking about the language, and most often refers to something "of Bangladeshi origin".
"Bengalese" is a combination of the root-word "Bengal" which comes from the Bengali language and the English suffix "-ese" which, in fact, is a borrowing from Old French, where the word-ending was "-eis" which in turn developed from Latin where it was "-ensem" or "-ensis" meaning "originating in" or "belonging to".
This word probably dates back to 19th century English literature where it has a similar meaning to "Bangladeshi", only referring to something of "Bangladeshi origin" and not the Bengali language.
Conclusion: Bengali, Bangla, Bangladeshi or Bengalese?
So which word should be used for which context?
Bangladeshi and Bengalese are both words that refer to people and things that come from or originate in the Bengal region in Northern India and Bangladesh. The two words don't refer to the language.
If I were to pick a favorite, I'd choose "Bangladeshi" because it's a word that originates from the region, whereas Bengalese is a European construct. (But why choose, though?)
As for Bengali and Bangla, they both have the same two definitions. One is "originating from Bengal" and the other is the language spoken by Bangladeshis.
If I were to pick one (and I suppose that I am), I'd go for "Bangla" because it's almost purely of the Bangla language, whereas Bengali actually is a Hindi word.
But again - never mind which word you use. The important point is that "Bengalese" and "Bangladeshi" are adjectives that only refer to something of Bengal origin, whereas "Bangla" and "Bengali" can mean the same thing, as well as being a noun meaning "language of Bengal".
And that's it.