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Some online websites and some books as well suggest that elements are either atoms (e.g. Ne) or molecules (e.g. $\ce{H2}$, $\ce{O2}$).
Original source: Tro, Chemistry: A Molecular Approach (2008)
Other sources say that only Fe and Cu are elements and $\ce{H2}$ and $\ce{O2}$ are not.
Source: https://slidetodoc.com/matter-matter-anything-that-has-mass-and-takes-5/
Can anyone explain and provide a reputable reference to the correct definition?
Here is one possible definition:
each of more than one hundred substances that cannot be chemically interconverted or broken down into simpler substances and are primary constituents of matter. Each element is distinguished by its atomic number, i.e. the number of protons in the nuclei of its atoms.
Source: Oxford dictionary
This starts with the classical definition (can not broken down into simpler substances) and then gives the modern picture of subatomic particles.
Here is the more technical definition:
1-A species of atoms; all atoms with the same number of protons in the atomic nucleus. 2-A pure chemical substance composed of atoms with the same number of protons in the atomic nucleus. Sometimes this concept is called the elementary substance as distinct from the chemical element as defined under 1, but mostly the term chemical element is used for both concepts.
Source: IUPAC Gold book, doi.org/10.1351/goldbook.C01022 as cited by the OP in the comments
So samples containing pure iron, pure argon, pure dioxygen ($\ce{O2}$) or ozone ($\ce{O3}$), or a perfect diamond are all elements. The bonding is different in all of these (or absent, as in argon), but all are considered elements. Among the examples, only dioxygen and ozone form molecules.
The first IUPAC definition (1-A) implies that you can e.g. say water is made up of the elements hydrogen and oxygen, even though water itself is a compound and not an element. The IUPAC definition also makes clear that you can use the term element either at the macroscopic level ("This sample is pure iron, an element") or at the particular level ("The helium atoms are diffusing through the membrane. Other elements, with larger atomic radii, do not diffuse readily").
[OP] Can an element be a single atom or a molecule of same atom?
Yes, element can refer to single atoms, to a molecule or, at the macroscopic level, to a sample made up of atoms grouped as molecules or not. The only requirement is that all the atoms in your set have the "same number of protons in the atomic nucleus".
Good question, there is an entire book dedicated to your question: What Is A Chemical Element? A Collection of Essays by Chemists, Philosophers, Historians, and Educators, 2020, published by the Oxford University Press.
The concept of a chemical element is foundational within the field of chemistry, but there is wide disagreement over its definition. Even the International Union for Pure and Applied Chemistry (IUPAC) claims two distinct definitions: a species of atoms versus one which identifies chemical elements with the simple substances bearing their names. The double definition of elements proposed by the International Union for Pure and Applied Chemistry contrasts an abstract meaning and an operational one.
You may wish to get hold of this book!
At one time, element was a substance that resisted further analysis, i.e., it was impossible to break it down into simpler substances. This implied classical chemical analysis. For example, if we chemically try to break down the Fe atom by heating in an ordinary laboratory, it will be an unsuccessful attempt. The substance after heating at laboratory temperatures followed by cooling will still behave the same.
IUPAC modernized this definition, by stating that A species of atoms; all atoms with the same number of protons in the atomic nucleus. It is a collective and a comparative label for a group of atoms.
As an example, if we isolated an argon atom from a bottle of argon gas, we could not conclude that it was an element without counting and comparing its protons with those from other atoms in the bottle. In that bottle, if we find that all atoms have 18 protons, we can label it as an argon gas, since each atom has 18 protons.
Now coming to your main question: What about Fe vs. $\ce{O2}$. Iron is clearly an element by elementary definitions. You can say that $\ce{O2}$ is a molecule formed by two identical chemical elements, called oxygen.
However, now in everyday parlance, we can say that many elements exist as molecules in nature but not as free atoms like Ne, Ar, Kr, Xe. For example the element sulfur exists as $\ce{S8}$, the element called oxygen exists as a diatomic molecule $\ce{O2}$. The element are oxygen and sulfur atoms but in nature they exist as molecules. If someone gives a solid yellow piece of sulfur in your hands, colloquially, we might say that we are holding the element sulfur, but in effect the actual referal is to individual sulfur atoms, regardless of the fact that the atoms are chemically bonded to each other.
