OK: that's a good start, but the carbonate ion is not considered organic despite containing carbon. The same goes for graphite and diamond (though those aren't molecules), and carbon dioxide is also often not considered organic.
I've also seen people claim that a molecule must have a C-C or C-H bond to be organic, but that disqualifies urea. Urea was one of the molecules used to disprove vitalism (see below), so if your definition of organic excludes it then that's a problem.
Historical interlude: The split between "organic" and "inorganic" molecules comes from the belief of early 19th-century chemists that organic molecules could only come from living organisms or be made from other organic molecules. This belief was called vitalism- that there was some unique "spark of life" that organic molecules had and others didn't. One way this was disproved was when Friedrich Wöhler synthesised urea from ammonia and a cyanate salt, both of which were thought of as inorganic.
Back to the definition argument, another possible definition I've heard is that an organic molecule has to contain carbon and either hydrogen, oxygen or nitrogen- but that makes things like Teflon or CFCs inorganic when I'd say they're organic.
The closest I've come to a systematic definition I'm happy with is "A molecular species containing carbon covalently bonded to either carbon, hydrogen, a halogen, or two non-metallic elements one of which is nitrogen or oxygen"- but that's an ugly one, and I'm sure there's some edge case which messes it up...