Rap is closer to rhythmic talking while singing is manipulating the voice to create a tone. They are both types of singing because in its barest form singing is just using the voice to create music. But the technique, and the artist's intention is what defines what type of singing it is classed as. It has nothing to do with word count, take this passage from a Judas Priest song as an example.
He's singing this section very quickly but he's clearly not rapping. He's singing, because he's using a singing technique, and singing is his intent. And I'm sure there are many rap songs that aren't very fast at all, but they are still considered rap. And I'm sure there are some very melodic rap songs, but they're still considered rap. I would have used a rap example, but I really don't know rap music well enough to... What an uncultured swine I am.