Did I make a major mistake?
No, non-structural screws are suitable for the shelves in the photo. 2×4s and wood screws support millions of shelves in garages around the world.
Should I get some stronger screws to attach the beams to the posts?
Only if you're paranoid, or there'll be extra-heavy stuff. Based on this calculator and some conservative assumptions, each screw will support 50 pounds after the glue fails.
I think the most vulnerable joint is between left-right beams and front-back beams. There's two screws going through face grain into end grain; this joint is weak in bending and is the only thing keeping the structure from racking forward. Possible improvements include attaching the plywood to the beams, adding metal straps, using pocket screws, or using longer (4-5") screws.
Your confusion comes from engineers having a more specific definition of "structure" than anyone else. To an engineer, "structural" means it supports loads as dictated by the relevant building code. Shelves also have loads, but nobody regulates the design of your homemade shelving.