An organometallic molecule is one that contains a metal-carbon bond. Organometallic chemistry is typically thought of as a subset of Inorganic Coordination chemistry, although it also plays an major role in Organic and Materials chemistries. Organometallics also play smaller but important roles in other branches of chemistry such as Biological and Analytical.
Chemists sometimes refer to particular molecules as organometallic even though these molecules lack the required metal-carbon bond. While this is formally incorrect, it is a convenient shorthand when the molecule has reaction chemistry that derives from or will lead to an organometallic molecule. Here is what the leading journal of the field, Organometallics, has to say about the matter in its Scope and Editorial Policy at the American Chemical Society:
For the purposes of this journal, an "organometallic" compound will be defined as one in which there is a bonding interaction (ionic or covalent, localized or delocalized) between one or more carbon atoms of an organic group or molecule and a main group, transition, lanthanide, or actinide metal atom (or atoms). Following longstanding tradition, organic derivatives of the metalloids (boron, silicon, germanium, arsenic, and tellurium) will be included in this definition. Furthermore, manuscripts dealing with metal-containing compounds which do not contain metal-carbon bonds will be considered as well if there is a close relationship between the subject matter and the principles and practice of organometallic chemistry. Such compounds may include, inter alia, representatives from the following classes: molecular metal hydrides; metal alkoxides, thiolates, amides, and phosphides; metal complexes containing organo-group 15 and 16 ligands; metal nitrosyls...."
Although most chemists refer to the field as Organometallic, some use other names either out of habit, to stress one part of it (the organic or the metallic), or to narrow the scope. Some examples include: