The definition of what's an organic compound is a little hard to define rigorously. Definitions have changed over the years, and at different levels of the student. Your best result would be to likely ask your instructor, write the answer down carefully, memorize it, regurgitate when needed on an examination, then forget it until the next time, and repeat the process.
Generally, we don't consider CO2, CO, or the salts of carbonic acid (example Na2CO3, CaCO3, etc) or carbides (example FeC crystals found in tough alloys of iron) to be organic, but all other carbon compounds are. Centuries ago, we considered the above chemicals as "rocks" or minerals and other "dead" things, and organic compounds as "alive." But that definition soon died as we understood organic compounds better.
Note: your definitions still fail 'tho. Carbonic acid has hydrogen, and is result of carbon dioxide in water and attacks minerals to produce the salts above, but are all defined as inorganic. Sure metal carbides are inorganic, but organometalic compounds are very important for organic synthetic reactions, and are definitely large organic molecules that happen to contain metal atoms. You wouldn't say the the iron containing hemoglobin, in your blood, is inorganic would you? You ... you're not a cyborg, are you?