Strictly speaking, is air homogeneous?
Strictly speaking you might have a point there. But it is probably more difficult to judge.

I was unable to unearth an official recommendation by the IUPAC, so I unfortunately have resort to Wikipedia ([1] Kristi Lew (2009). Acids and Bases. Essential Chemistry. New York: Chelsea House Publishing.):


  Homogeneous is a term in physical chemistry and material science that refers to substances and mixtures which are in a single phase. This is in contrast to a substance that is heterogeneous.[1] The definition of homogeneous strongly depends on the context used. Generally it refers to smooth variation of properties with no discontinuities or jumps.  


If we talk about all the air around the world, it is certainly not uniform. There are concentration differences across certain areas. When we consider a vector into space, the total concentration will decrease along this vector. As we know, we will pass certain areas with different chemical compositions, for example the ozone layer.
But all these changes will be gradually; there are no discontinuities. In this sense one would call air homogeneous, even though it does not fit in the definition of phase.  

Additionally, if you take an arbitrary (volume) unit of air, all at the same temperature and pressure, and you allow it to establish an equilibrated state, then yes, this will not only be homogeneous, it will also form one phase.
The above linked article further considers air as a homogeneous mixture. It rambles on with considering it a gaseous solution of oxygen, carbon dioxide, etc. in nitrogen. This is technically incorrect, since the gaseous state of matter is specifically excluded from the IUPAC definition of solution. Also a solution can never be at the same time a mixture, as the former can be separated by mechanical means, while the latter cannot.

The problem here is obviously the lack of a unique and rigorous definition for homogeneity. For whatever purpose when it really matters, one should probably use a more robust definition like phase, which is essentially what your second statement describes.



Depending on the context you might even obtain a homogeneous mixture of water and oil - the question is then more or less how stable such a suspension is. After a while it will certainly separate in two phases.