Learn English – the “hole” in this desk called

I think both can be used interchangeably. However in my view, authors write books while writer is a more generic term. The are other words for other media too, playwrights write plays, bloggers write blogs, reviews write reviews, poets write poems, but all are writers.

The three definitions overlap substantially, but usage of the three differ by context.

Let's put the three words into full sentences:

1. This piece of paper was cut.
2. This shape was cut out from the page.
3. A corner of the page was cut off.

Sentence 1 just means that a sharp tool was applied to the paper, causing one or more incisions. (E.g. a rectangular piece of paper can be cut into the shape of a star, an apple can be cut into quarters.) There is often an expectation that the cut starts from an external part of the original unless the phrase is expanded to cut out. You can say that the left and right figures were cut; a weaker case can be made for the middle figure.

Sentence 2 indicates the relationship between the shape and its 'parent' page. The word "out" gives the phrase "cut out" the notion that the shape was cut from the page's interior region. This closely matches the middle figure.

Sentence 3 stresses the break in connection between the corner and the page. "Cut off" normally refers to the removal of a protruding section. This matches the figure on the left best.