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By far, my understanding is that a molecule is made up of atoms bonded together. For example, a molecule of water ($\ce{H2O}$) has 2 hydrogen atoms and one oxygen atom.

However, when it comes to Avogadro's number, I'm getting confused because it mixes the concept and treats it like it is the same.

Avogadro's number is defined as the number of elementary particles (molecules, atoms, compounds, etc.) per mole of a substance.

If I take 1 mole of water, with the molecules definition for the Avogadro's number, it has $ \mathrm{6.022×10^{23}} $ molecules of water. With each molecule having 3 atoms, it means that 1 mole of water has $\mathrm{18.066×10^{23}} $ atoms.

If I take 1 mole of water with the atoms definition for Avogadro's number, it has $\mathrm{6.022×10^{23}}$ atoms of water.

What is the right way to think about this?

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    $\begingroup$ A mole is always just a number. Out of context that means you don't know what you are counting. You always need to define exactly what is being counted. This resolves the ambiguity (one mole of water molecules contains one mole of oxygen atoms but also two moles of hydrogen atoms: there is no ambiguity if the thing being counted is specified.) $\endgroup$ – matt_black Mar 5 at 18:34
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    $\begingroup$ The Avogadro number is just a number, like 12. You can have 12 eggs, or 12 boxes of eggs. $\endgroup$ – Poutnik Mar 6 at 5:04
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$1$ mole of water has $\pu{6.022E23}$ molecules of water, but not $\pu{6.022E23}$ atoms of water, because the expression "atoms of water" has no sense. You are allowed to state that it contains $\pu{18.066E23}$ atoms. But you are not allowed to speak of "atoms of water". Water has not its own atoms of water.

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