I have found the following definition on Wikipedia:
An organic chemical compound (or related set of compounds) is called a vitamin when the organism cannot synthesize the compound in sufficient quantities, and it must be obtained through the diet;
That doesn't seem to exclude essential amino acids and essential fatty acids though, and those are also organic chemical compounds that by definition can't be synthesized in sufficient amounts.
So is the definition incomplete?
Generally speaking, bulk essential nutrients (e.g. amino acids, fatty acids) are used stoichiometrically to form macro cellular structures and are 'baked in' to the structures or compounds they are used to make. Once those structures are destined for degradation, sometimes the bulk essential nutrients can be recovered and reused in the synthesis of other structures or compounds, but catabolic elimination from the organism is always an option that yields useful energy.
Vitamins, on the other hand, are usually used catalytically. Vitamins are not 'baked into' the compounds they help make or process; vitamins transiently participate in the reactions required to make or regulate a compound or structure, but don't ultimately get 'used up' by that process.
Ultimately, vitamins don't have a bulk energetic value to a cell. For instance, once vitamin E (α-tocopherol) is oxidized, its only fate is to be excreted as waste. Fatty acids and amino acids, once destined for degradation, have their remaining chemical energy extracted prior to excretion of their downstream waste products.
It is philosophically interesting to remark on the diversity of bulk biomolecular building blocks that are still necessary for some organisms to obtain from their environment. Not all of them get called vitamins.
It's just naming convention. We recognize, officially, the 13 vitamins based on activity, but we intentionally classify amino acids, etc. as such instead of as vitamins. It's just one of those things that people stick to so that it's standard, such as beverages as a blanket term for certain drinks but not for every type of fluid. (maybe that's a bad example, but hopefully you get the idea)
