Mr.Parker - I would like to state again for your benefit and hope you are able to follow this time - Yes! Raimi's Spider-Man was based on origin story of Spider-Man, but with some changes and organic webbing was one of them, based on success of this movie, MARVEL comics accepted organic webbing as power for Spider-Man and after 2004 (after release of movies) also had published stories in comics namely - 'Queen' & 'The Other' - where Peter Parker undergoes further evolution to gain organic webshooters and more powers...
Though 'Queen' wasnt a great evolution story and 'Other' was better, my point is MARVEL COMICS have, though late, accepted organic webbing as power for Peter. so organic webbing are true to comics.
Btw in ASM except for webshooter, I dont see anything close to the comic story, this is anyway more like ULTIMATE series of comics than the original story by Stan Lee in 1960s... so there....