"Shut the door" is not a term used in the rules, but rather a convenient metaphor (probably from the days of door-to-door salesmen -- you had to shut it before they got a "foot in the door"). What I mean by it is that L, when she changes course to the point where there is no room for W to pass between her and the mark/committee boat, at that time W must have an opportunity to avoid the mark (by slowing down or passing it on the other side. If so (and L thereafter holds her course), then L has "shut the door" on W. If L waits until W can no longer avoid the mark by slowing down or turning away before changing course, then L has "let her in".

When talking to sailors about starting tactics, I often advise that they take a transit on the layline to the signal boat. That way, they know where to turn close-hauled in order to shut out windward boats (without getting shut out themselves). Provided you don't let them get clear ahead of you, you can "close the door" on windward boats well back of the signal boat.

I hope that helps,
Eric