Usage of colon in English writing
Solution 1:
I believe your use is fine in both instances, and your understanding of its purpose is correct. Especially in example one, I think a colon is an excellent way to point out the main goal of an academic paper.
Note: Some (but not all) style guides call for the clause following the colon to begin with a capital letter if it's a complete sentence (as in example two).
Solution 2:
Your use of the colon in those two sentences is totally standard. It is indeed equivalent to ", which is:" and this is OK.