I think that mental model only works if you imagine the parabolas as reaching to infinity in a finite space so that both ends are parallel, ie having identical vertical slopes of +/- infinity. At that point, easier just to call it “half an ellipse”. To me, it’s much easier to imagine a parabola as the end of an infinitely long ellipse.

Your intuition and the KSP example are correct though. If you imagine the plane and cone for a parabola, you wouldn’t notice any significant change to the shape (at a finite distance) if you tipped the plane ever so slightly into forming an ellipse (or a hyperbola, for that matter) since it’s all smooth changes.

Anyway, the size of the elliptical (I think hyperbolic would have a different sort of energy state) arc that’d be formed by a thrown object would be so large relative to human scale as to basically be infinite, equivalent to a parabola. I imagine the difference might become significant once you are launching something a decent way around the Earth, but with that much energy in play I don’t think it makes much difference where exactly the projectile “lands”.