Your Snoopy & Woodstock Handcar was produced in 1990, and was the first to use a revised chassis. Therefore, the link and directions that are provided won't work.
Here's the link for the revised Handcar. Larry's instructions about removing the pin from the handle still apply, but that's where the similarities end. Instead of a screw holding the body to a stamped metal chassis, there are four tabs on the plastic chassis holding it to the body. You can see the tabs in each of the four corners in the diagram. Pull outward on one of the body's "bumpers" slightly to disengage the tabs and separate the chassis from the body. (Everything is fairly robust, so I wouldn't be too concerned about breaking anything.)
I hope this helps.
TRW