How do we know that nuclear physics corresponds to low energy QCD? One often hears the phrase "most of nuclear physics is in the low energy regime of QCD, where strong coupling constant is large ...", with the following diagram being invoked:

How does one know where nuclear physics fall on the x-axis of the graph?
If it were related to the energy scale of the quark bound states like pions, if I have a counter-argument. Yes, the mass of a pion is around 140 MeV, but another example of a quark bound state is a proton, 939 MeV, way higher up on the x-axis. Objects like the Xi hyperons are still more massive. This way, you can go higher and higher towards the right, on the x-axis in that graph. Clearly, this logic is flawed.
So, what is the real logic behind insisting that nuclear physics and low energy QCD correspond to each other?
 A: A link would be useful to see the context of the quote

One often hears the phrase "most of nuclear physics is in the low energy regime of QCD, where strong coupling constant is large ...", 

As it is it is wrong. This is correct 

The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves

Note the word "residual" which is missing from your quote. It is not enough to talk of low energy regimes of QCD. 
Like Van der Waals forces between neutral atoms, the nuclear force is the spill over force coming from strong interactions between colour neutral nucleons.
QCD has three charges , labeled with color indices,  and a proton or a neutron are color neutral. There are spill overs from the strong gluon exchanges that bind the nucleons that generate the attraction and known as nuclear force which make protons and neutrons attract each other and build the periodic table of elements. These are of order MeV and distances of fermi, as determined experimentally.


Straight lines are quarks, while multi-colored loops are gluons (the carriers of the fundamental force). Other gluons, which bind together the proton, neutron, and pion "in-flight," are not shown.

In this image we see  a possible diagram of the attraction between nucleons, which generates a pion exchange in keeping color neutral at the "large" nucleon distances. The binding between the quarks in the nucleon is much stronger than the nuclear residual attractive force at the energies of MeV. It needs very high energies to drastically intervene in the structure of nucleons, as happens currently at the LHC.
For nuclear energies the plot/calculations  have been simplified as a Feynman diagram with a pion exchange.

