For "statement" and "proposition", the terms can get jumbled depending on how each term is being defined, but the idea is clear and builds on this distinction:
- There are expressions in languages (Schnee ist weiss, Snow is white, 雪は白い)
- There are logical meanings and relations (S has property W)
1 and 2 are not identical. The former have to follow the rules of their natural languages and can be vague or ambiguous. In the English version, "white" could be either a noun or an adjective and this would change its logical meaning.
Conversely, propositions in logic are intended to express 1 thing precisely and to be trans-lingual entities. No proof should ever hinge on multiple senses of a word or on a vagary of a particular language.
Logical proofs (especially formal ones but in a less strict way most philosophical proofs) deal in 2. But depending on how formal the area is, we still write in natural languages (with the goal of being precise enough about our terms to prove things).
To give an analogy, the letter c can express two sounds, which we write as /k/ and /s/ but the /k/ and /s/ are not letters, they are IPA ways of writing the pronunciation. Propositions are that IPA level even if we write them as if they were natural language.
While some people use "statement" to mean what others use "proposition" or "claim", they are all angling at roughly the same idea above.
None of the people involved would mean a "declarative sentence" by "statement" because "declarative sentence" is a term of grammar -- and merely refers to the form of a sentence. What people are looking for in logic is not an understanding of grammar but an understanding of the logical meaning and how it is organized. (I can vaguely imagine someone rejecting the term "statement" because they view it as a synonym for the grammatical term "declarative sentence")
This is a big part of Russell's work.
Of course, there are philosophers who work on the philosophy of language who are interested in the relation between natural language and logic, and there are other philosophers who think the idea that there's a logic meta-language that differs from natural sentences is bunk.