I don't think Marvel even has a clear idea of what the "Spider-Family" is beyond a name that can encapsulate all the disparate Spider-Characters .
I think your understanding of the concept is a good way of looking at it, and Peter's even referred to them as his "web," but I think in practice it's more difficult because of the idea of multiple spider-people existing and sometimes interacting in the same city taking away from the individual and independent aspect of Spider-Man, especially when big storylines in ASM often get dedicated to or focused on the other spiders.
I think it's also an issue for other media that tries to incorporate multiple spiders in one setting in a way they weren't designed to be since, as you alluded to, they are meant to be more separate and uninvolved then they are in the cartoons or movies.
Just look at how badly poor Miles gets treated in the Peter-led Spider-Man cartoons he's in. Reduced to a sidekick in both and saddled with a terrible name (Kid-Arachnid) in one or no codename on the other (Marvel's Spider-Man). At least in the comics he gets to be his own man and hero, if not as much as he was in his own universe where he was pretty much the sole Spider-Man.