The movie was grammatically incorrect. That's not uncommon. Many people don't know the difference (or even that there is one).

Lie implies that you yourself are lying down. "I want to lie down" as you said. To lay implies you're laying someone or something down. "I am going to lay the baby down so she can sleep." for example. Or, "I am going to lay the book down on the table."

It's a bit pedantic to nitpick it, though, so I seldom do. Many books, songs, movies, etc incorrectly use one or the other. For example, the song... I don't remember the title, but it goes something like... "If I lay here, if I just lay here, would you lie with me..." anyway. It uses them correctly sometimes, incorrectly others.

Don't stress over it. Either way, people will know what you mean.

Don't even get me started on past tense of each, and past participle. (lay, laid, lain, etc)