In OP's first example...
1a: Think about the movies you like to watch or the books you like to read
...the highlighted definite articles are completely optional, and most people would probably say that...
1b: Think about movies you like to watch or books you like to read
...means exactly the same thing. But if I had to make a distinction (which I wouldn't normally want to do), it's that the "article-less" version #1b could be thought of as talking about the kind of movies / books that you like to watch / read, whereas #1a is explicitly referring to the particular, specific movies / books that you like.
The second example is different...
2: Is there any way to count the number of movies one has seen?
To be grammatical, this one must include an article. In principle it could actually be the indefinite article (a number of movies), but in practice for the exact context as given, we just wouldn't do that, since there's definitely only one specific number being referenced. Even though we don't know the value of that number, it's contextually relevant that it has a specific value.
On the other hand, in a context such as...
3: I have seen a number of movies
...we always use the indefinite article. Note that this one is an idiomatic usage, where a number [of X's] actually means quite a lot (not very many, but definitely more than a few).
In which context it's also worth noting that quite a few has the same meaning. I imagine it might seem odd to some native speakers that quite a few usually implies a lot, whereas plain a few means not many, and quite a lot usually implies more than "a few", but less than "a lot".