"page 2" or "Page 2"?

There is a difference between "page 2" and "Section 2". Wherever the "Section 2" is located, the author most likely has actually given it the title "Section 2", so that is the name of a specific section of your paper. However, "page 2" is a page with the name "2" (not the name "Page 2", unless you wrote "Page #" in the corner of every page). This, I believe, is the logic for writing "page 2" in lowercase. It is like the difference between "Bob's house" and "White House".

However, the real answer actually depends on the author and the audience (in a certain field, people might expect publications to give pages as e.g. "Page 2" out of tradition or whatever reason, in which case an author would follow that).