Line is used in a number of different ways as is indicated by the various answers to this topic.
The question: "whose line are they from" can be a conversation starter about your birds, a querry about their background from a prospective purchaser, or a question to gather how much knowledge and experience you have as a breeder.
Some lines exhibit slightly different traits than others, but all within the standard. I recall reading a post on the silkie group years ago about a breeder who had bred into his birds (sorry, I don't recall which of the big names from 8-10 years ago) very thick feathering--more feathers per sq inch. Either the same or a different breeder had bred in crests that were large, but the areas in front of the eyes were not, so despite their larger crests, their vision was not blocked. These traits are acquired by selective breeding over many generations.
Some breeders try to keep their birds at the small end of the scale, while others aim for the top end. I know one breeder who selects for longer feather length.
As for line breeding, there are ways to use line breeding that emphasize genetic diversity while still concentrating specific traits. I think Laura Haggerty has an article on her website on using three lines for breeding. With less space you can use two lines to a similar purpose. Of course that does not provide as much diversity, but it provides more than using a single line.
When one uses the name of another breeder in reference to one's own birds, it should be used in a complimentary fashion, giving them credit for the quality of their birds. If you don't have something good to say about a breeder, don't say it.
It is one thing to say "my birds have Stanford, Mahalik and other top breeders in their background," but quite another to say "100% xxx" when xxx did not breed them, even if the parents are both 100% xxx breeding. As was stated, xxx did not make the pairing decisions.
When do the offspring become your own line? Hard to say. It's one of those things that develop slowly until one day you say...my line, and realize it has been so for quite a while.