What to say 'one mole of C atoms or one mole of C molecules?
What you need to remember when using terminology involving moles is that moles are just a way of counting things but that the terminology only makes sense if you unambiguously specify the thing you are counting. It is entirely possible to use moles to count non chemical things (for an amusing example of the consequences of having a mole of moles (the small burrowing mammals) see this.) Most confusion arises because the way the thing is specified is ambiguous.

Some examples:


a mole of CO molecules is perfectly correct (the units being counted are single molecules of CO)
A mole of C atoms is perfectly correct (and, if you are counting C atoms in CO molecules that is also fine and the answer will be the same)
a mole of C molecules in CO isn't correct but only because the molecules is CO which consists of atoms of C and O.
if you were counting the atoms in CO2, then it would be correct to say there are twice as many moles of oxygen atoms than of carbon atoms or carbon dioxide molecules


The use of moles to count doesn't imply anything about what you are counting, you have to specify that as clearly as possible. When talking about the parts of a molecule we usually talk about the atoms that make up the molecule. CO2 doesn't consist of molecules of oxygen and carbon, but atoms of oxygen and carbon. But this is independent of the use of moles as a way of counting those components.