The article overstates the obvious, and is attempting to offer advice on a very complex subject in 500 words or less.
As others mentioned, in some companies the only way to get more money is to leave, or threaten to leave.
Of course you have to consider very carefully taking a counteroffer these days -- I'd even go out on a limb and say you have to consider leaving and taking a new job more carefully now than in the past.
The situation is different in each case.
A counteroffer could offer more than just money -- perhaps it involves another position or different duties. Perhaps your notice put the company on notice that others might be unhappy in your departement.
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