Quarkonium: a concept for heavy quark-antiquark pairs, not for light ones? I am concerned about the concept of quarkonium. Actually, it is for heavy $q\bar{q}$ pairs (of the same kind) bound by the strong interaction. My question is: Why don't they include the light $q\bar{q}$ ones (u,d,s) in this concept? 
What is the difference between those two types of $q\bar{q}$ pairs?
 A: The comment by Dukwon holds . It is only heavy quarks that make -oniums.
It seems to be a term useful in lattice QCD calculations where fine structure effects can be calculated and compared to experiment

"Charmonium" is a bound state of a charm quark and anticharm quark. The term "quarkonium" refers to a bound state of any heavy quark and its antiquark. These states are unusual because the quark masses are much larger than the typical confinement scale. As a consequence they have low-lying excited states quite analogous to "positronium", a bound state of an electron and positron, from which the name is derived. The masses and decays of low lying states of charmonium have been quite accurately measured and allow an exacting test of lattice QCD methods.

italics mine.
So it is the analogy with positronium states that makes the quarkonium states calculable in lattice QCD.
The low mass quarks do give excitations, (eta, eta'), the mass differences are large and they are treated in a different manner in lattice QCD.
In this table, there are no I=0 low mass mesons. Up antiup by itself does not make a distinct resonance . The phi is an s s_bar resonance. In the link it is argued why it is most probably a pure s s-bar, and it has to do with the quark  mass mixing. It is even treated as a quarkonium in some references. Note equations 6 and 7, where a gamma accompanies a quarkonium change of energy level ( as in positronium energy levels)
