The "search" fields should exclude movies that don't match (from the list).
"Match" should mean that the title (and optionally all text fields - checkbox?) contain the characters (case insensitive).
So for example typing "t" would eliminate "A bugs life" from the list, typing "r" would
eliminate more and by the time I got to "tru" or "truman", I'd be left with "the Truman Show", "truman show" etc (and True lies").
I'd say multiple words should simple mean that all those space separated character sequences were found (not one after the other), perhaps quoted to mean that specific sequence of characters.
This would show up all matching movies whether thay started with "the" or not as well as allowing you to remember that it was about "zombie" and find matches that way because the description contains that word.