Zinnser makes a variety of primers, including alkyds, water-based (latex), and oil-based. You have to use one that is best for the surface.
The type of primer also has to match the type of topcoat. Vinyl surfaces are best coated with 100% acrylic latex paint. Latex primer, latex topcoat. Flat or satin finish preferred, semi-gloss at an extreme. No high-gloss.
As long as your primer coat was adhering well enough, the surface is probably rough enough for good paint adhesion, so reapplying the correct primer (after removing the previous primer) shouldn't be a problem. If the primer didn't adhere well enough, after removing the old primer, roughen the surface a little bit by scratching it with a medium-grit sandpaper before applying new primer.
Unless you want the topcoat to be bright white, follow this handy hint: start with a white primer coat, and make each successive top coat darker than the one before until you get to the topcoat color you want. Not only do you get less bleed-through that way, but you can tell at a glance if you missed a spot. If you want the finish coat to be white, then ALL coats have to be white.
Engineering: Finding complex solutions to simple problems you didn't even know you had.