When I was a child, I thought I knew what a dinosaur was. But, as I grew older, I was told that many of the extinct species that I thought were dinosaurs (dimetrodons, pterosaurs, plesiosaurs, etc.) were not in fact dinosaurs. Is there a reasonably quick way for a lay person to tell apart true dinosaurs from non-dinosaurs without taking up an in-depth, species-by-species study of paleontology?
There are several defining morphological characteristics of dinosaurs. Including their ankle structure and perforated acetabulum, Aka the hole in their hip goes all the way through, as well as a few other minor features. These combined makes it literally impossibly for a dinosaur to take on a sprawling pose without breaking hind leg in many places, this is why you will often see an erect posture given as the reason, but that is a product of these feature not the defining feature itself, there are many ways to achieve upright posture.
Now of course we are moving away from purely morphological definitions, cladistically dinosaurs are defined as The last (most recent) common ancestor of Triceratops horridus and Passer domesticus (common pigeon), and sometimes Diplodocus carnegii as a third, and all of its(the ancestor's) descendants.
A more superficial quick and dirty method is if it is a pterosaur or lives in the ocean it is not a dinosaur, If its limbs sprawl it is also not a dinosaur. I should also include this guys work, since he attempted to create a complete cladogram of every dinosaur, so you can visualize the group.
Yes, the legs of dinosaurs are directly beneath their bodies. This is more efficient that the sprawling stance of lizards, crocodilians, pelycosaurs, and other mammal-like reptiles. It is likely no coincidence that another successful group, mammals, also have their legs directly beneath them. Also, as far as I know there were no fully marine dinosaurs.