Both of these definitions of buffer solution are given by the authors of chemistry textbook.
In the first definition, Authors says " You can treat buffer solutions quantitatively if you use the appropriate Ka or Kb and recognize the role of common ion".
What does it mean? What message authors want to give to the readers? Would anyone elaborate on this topic?
He is probably referring to the Henderson-Hasselbalch equation which is pH = pKa + log([A-
Another version exists where you use pKb instead, and as you know, pKa and pKb depend on Ka and Kb. You also derive the HH equation from Ka or Kb.
If it's not that, I don't know what he's talking about.
As for the common ion effect, recall that the concentration ratio uses equilibrium concentrations. When calculating such concentrations (like with ICE tables), one must take into account the common ion effect.