Note that the square root term is the standard deviation of the mean difference. You can check that this expression reduces to Expression 1a when n1 = n2.
This issue has been discussed at length by Leventhal and Huynh (1996). They argue for a directional version of the null hypothesis, but this differs little from standard practice based on visual inspection. Their main finding is that standard power calculations yield slight overestimates in selected extreme cases.
In the range view of the null hypothesis, Type I error includes Type III error as a special case. If the true effect is substantial, on the other hand, a wrong directional conclusion is improbable.
But we may have 50% confidence in heads; even odds on the outcome is a fair bet. Confidence is thus a valid guide to action (see Confidence, Probability, and Belief, Section 19.1.2, page 607).