I suggested several ways of resolving the factor of 2 discrepancy in the previous post, but one thing I did not suggest in that section was what I used to start the post: Allow for the possibility of the beam being simultaneously compressed and bent.
The intuitive way I understand beam buckling:
Put force on a beam. It compresses (linearly). It takes a certain amount of energy to compress it linearly. At some point, it might have so much energy stored in compression that it may be able to relieve some by returning to a longer length, but bent. At that point, the beam has buckled.