Actually pulling the break material out of solution rapidly is what matters, but whether it goes into the fermenter or not, really doesn't. If you pull the break it just settles to the trub with everything else, and doesn't affect the taste or clarity of your beer. It is the mechanics of getting those proteins out of solution that really matters. Think of it as if you dropped some vinegar into a glass of milk, how some stuff comes out of the solution and curdles (the whey and curds break apart) that is what we're talking about with "break" material..Certain proteins actually breaking down out of the wort. Get it? "Break?"
But whether you leave it behind or put it in the fermenter? It really doesn't matter either way.
Some dump everything in, without straining, just pour it in the bucket or in the funnel....Some use a big strainer that fit in the funnel for a carboy, or a sanitized 5 gallon nylon paint strainer bag in the bucket...
I have done it all ways. It really doesn't matter...anything will settle, my beer is plenty clear either way.
In other words, there is no wrong way to do it, or better way, or way that will make the best beer...they all work...the choice is what will work the best for you. That's how you develop you own unique brewing process. By trying all ways and deciding what works best for you.
What I do with my IC, is chill the wort, then I lean the bottom of my autosiphon about two coils up from the bottom on the metal of the siphon. That rests it above most of the break material and trub, then I rack it to the fermenter until I'm down to that and carefully lower the siphon down into the gunk, just trying to get as much of the wort as possible without letting in the hops and break matter.
But pretty much up until I got my immersion chiller for christmas last year I just dumped for the majority of my batches.