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According to the definition of organic compounds on Wikipedia:

An organic compound is any member of a large class of gaseous, liquid, or solid chemical compounds whose molecules contain carbon.

Carbonates, simple oxides of carbon, and cyanide molecules do have carbon, but they are not regarded as organic. The same Wikipedia article on organic compounds reads:

The modern meaning of "organic compound" is any one of them that contains a significant amount of carbon.

The current main definition for organic compounds doesn't fit for carbonates, simple oxides of carbon, steel, etc. So, what should be the redefined definition for organic compounds?

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    $\begingroup$ The next sentence in the Wikipedia page you cite is “The distinction between "organic" and "inorganic" carbon compounds, while "useful in organizing the vast subject of chemistry... is somewhat arbitrary” $\endgroup$
    – F'x
    Commented Oct 29, 2013 at 17:42

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So, what should be the redefined definition for organic compounds?

We should not define organic compounds. As F'x pointed out in the comments, the distinction between organic and inorganic is largely arbitrary and mostly based on a distortion of historical precedent.

Rather, we should classify compounds by the type(s) of bonding interaction present in the substance. By this criterion, we have several classes of compounds:

  1. Molecular (or covalent) compounds - the particles that make up these compounds are molecules held together by covalent bonds. This class of compound includes traditionally inorganic $\ce{CO2}$ and organic $\ce{CH4}$ carbon compounds. Covalent compounds also the majority of organometallic compounds, so we do not need a separate category for these. Other traditionally "inorganic" compounds, like glass, also belong in this category.
  2. Ionic compounds - the particles that make up these compounds are largely ions held together by Coulombic forces. For the sake of simplicity most people would include in this class of compounds those ionic compounds where one or both ions is a small molecular ion held together by covalent bonds $($e.g. the $\ce{NH4^+}$ ion in $\ce{NH4Cl}$ or the $\ce{CO3^{2-}}$ ion in $\ce{Na2CO3})$.
  3. Mixed covalent/ionic compounds - those in which there are some ionic functional groups, but mostly covalent bonds, like methyl orange. Many folks might list these substances in Category 1. This category would include most coordination compounds including such things as metal-organic frameworks and probably many composite and/or ceramic materials.
  4. Metallic compounds - those substances with extremely delocalized bonding, typical of metals. Alloys, including steel, belong in this category.

Under this system, what we traditionally study as organic chemistry becomes redefined as the study of the structure, properties, and reactions of covalent compounds (Classes 1 and 3), especially the nature of the covalent bond and how it can be formed and broken selectively. What you might learn about the structure, properties, and reactions of organic compounds can easily be extended to the structure, properties, and reactions of molecular inorganic compounds. For example, we do not need two different branches of chemistry to study benzene and borazine.

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The term "organic" in chemistry historically means "requiring a living organism for its creation", as proposed by the vitalism theory. The modern definition differs and can vary, as the answers to posts on this site demonstrate. It might help to quote IUPAC's definition, curiously found under the definition of "functional group":

Organic compounds are thought of as consisting of a relatively unreactive backbone, for example a chain of $\ce{sp3}$ hybridized carbon atoms, and one or several functional groups. The functional group is an atom, or a group of atoms that has similar chemical properties whenever it occurs in different compounds. It defines the characteristic physical and chemical properties of families of organic compounds.

This fits the definition provided by Ben Norris in this post, but disagrees with that by tomij, which requires the presence of atoms of covalently bonded carbon in the compound, although that definition is common.

References

(1) IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. https://doi.org/10.1351/goldbook.

(2) PAC, 1994, 66, 1077. (Glossary of terms used in physical organic chemistry (IUPAC Recommendations 1994)) on page 1116.

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