Electrode voltage curves (and the combined Cell open-circuit voltage) are steeper when the electrode is delithiated. Therefore, during discharge, delithiated anode is the electrode which drives the OCV to the cutoff threshold. During charge, delithiated cathode is the electode which drives the OCV to the cutoff value.
After Loss of lithium inventory, anode and cathode are delithiated (during discharge and charge, respectively) slightly deeper than in a pristine cell until the change of the electrode's voltage compensates for the lower (or higher) voltage of the opposite electrode which is farther from being fully lithiated than in a pristine cell. The authors of  call this effect stoichiometric offset. BMS could prevent this deeper delithiation (which presumably accelerates degradation further) by changing cutoff voltage thresholds as the cell degrades.
On the left chart, the bars are aligned so that any horizontal line which crosses both bars represents a possible stoichiometric state of the cell: the sum of stoichiometry on anode (increasing from the bottom to the top) and cathode (increasing from the top to the bottom) is constant and is less than in a pristine cell.
The chart on the right (with negative electrode voltage curve shifted to the left) can be misleading because the X-axis scale (State-of-charge) is not well defined after cell has lost so much lithium inventory. State-of-charge is a relatively synthetic state abstraction which is estimated (e. g. using Kalman filter) from the Cell terminal voltage and Cell open-circuit voltage function (usually deemed static), hence SoC keeps varying between 0% and 100% regardless of how much capacity the cell has lost.
One of the reasons why electrode voltage curves are steeper when the electrode has little lithium is the physical underpinnings of these open-circuit voltage curves: see Electrode lithiation stages.
 Degradation diagnostics for lithium ion cells