The organic webshooter and the Spinnerets from the wrist actually came from Spider-Man 2099 where Miguel O' Hara grows them on his forearms and even jokes that at least they aren't shooting out of his butt. The idea for the "finger talons" were also borrowed from there as a suggestion of how he sticks to things like a Spider although less dramatic in the Raimi version:
The spider falls from the web and bites Peter, he grabs the web and yanks the actual Spider off as it has embedded itself via it's fangs at that spot. The wound is of a burrowing spot which some Spiders do to their prey.
The webshooters are 100% non-organic as evidenced further by him testing it by shooting the web through the wreath and it being completely unattached to him. If the wound on the back of the neck were a spinneret why would he need a webshooter? Not to mention he'd need a way to lace it from his neck to those webshooters on the wrist and it'd be the wrong part of the back, the Spinneret would be placed at the tailbone area of a human.
While we're on the topic, watching him turn the dial and change the intensity of the web made me smile as it hasn't really been done since the Bronze age were they showed that depending on the pressure and how he handled the trigger, the web length, intensity, even making those "Web balls" were possible. I remember that trading card with the info on it and even the full page in their old "bibles" that the details thought up.