Statistical Significance

Not really.

No. Clearly those equations don't back up what fls said on the other thread - that "smaller effects increase the likelihood that that positive results are false-positives, even in the setting of very low p-values". If you keep the p value the same and reduce the effect size, then the false-positive probability can either go up, or go down, or stay the same. It depends on the power. If the power is the same, the false-positive probability is the same.

Nor, in my view, do they back up what fls said on this thread about the importance of the power and the unimportance of the p value. Clearly, they both play a role.

Anyway, the virtue of simple equations like those is that they are quite simple to interpret. They are far more efficient than verbal descriptions in a situation like this.