Atoms, molecules and stoichiometry: describe question

Water is sold in bottles, and the number of water molecules in one bottle is far too large to be counted individually. A manufacturer records the amount of water used in each batch in moles. The Avogadro constant is $6.02 \times 10^{23}\,\text{mol}^{-1}$.

A label on a reagent bottle reads '1 mol oxygen'. A student says that this label could describe two different quantities.

Describe what is meant by one mole of a substance, how the Avogadro constant links the amount in moles to the number of particles present, why the student is correct, and why chemists use the mole rather than counting particles directly.