the book and pen == instead of == the book and the pen
These do not necessarily mean exactly the same thing. If pen is part of a set with the book (e.g. A diary with a builtin loop to which the pen is attached) it would be more natural to say hand me the book and pen rather than hand me the book and the pen.
- Suppose we have a non-mathematical example that does preserve meaning:
Some examples don't preserve meaning, but some do.
This sentence has two examples of ellipsis, in which a word or phrase is omitted and the meaning is implied.
Some examples don't preserve meaning, but some [examples] do [preserve meaning].
Omission of [examples] is noun ellipsis.
Omission of [preserve meaning] is verb phrase ellipsis.
- I would not use factor out to describe what you did in the black/white ball example.
"The black and white ball" refers to one ball that has two colors.
"The black ball and white ball" refers to two different balls, each of which has one color.
"The black and white balls", is ambiguous. It could refer to either multiple balls, each of is either black OR white, or it could refer to multiple bi-colored balls. Without additional context, I would assume it meant the latter (two or more balls, each of which is both black and white.)
Suppose context makes it clear that this is not the case (e.g. a table has three balls on it, each of a different color, and the magician instructs the volunteer to "pick up the black [balls] and [the] white balls"). This is noun ellipsis.
You are looking for a "non-technical" synonym for ellipsis. I can't think of one, probably because this type of analysis of sentence structure rarely happens outside of a "technical" context.