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.

$$1$$ 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.