Ultimately, Socrates's argument does not rely on (I) = (II). Rather, it relies on the difference between (II) and what I'm going to call (III):
III) Something is pious, because all of the gods love/admire it.
We cannot have both (II) and (III), because that would be circular (the pious is defined as that which the gods love, and the gods love it because it is pious). So if (I) is a coherent position at all, it can only mean one or the other of (II) or (III), not both. Socrates then goes on to criticize both (II) and (III) as problematic in their own right. (III) is a poor definition, in Socrates's reckoning, because Socrates holds that a proper definition should be in terms of a thing's "essence" (or essential properties) rather than its accidental relationships with other things (such as the gods). (II) is problematic because it does not actually define what "piety" is, and that was the point of the whole exercise in the first place. Socrates argues that we can't complete the definition without eventually appealing to the gods and their values, and then we're back to (III) again.
The emphasis on "essence" will sound strange in the context of modern philosophy, but it was considered very important at the time. By modern standards, the problem with (III) is that it is arbitrary - the god(s) may love whatever they wish, and who is to say that the gods are right about that? If we appeal to some external notion of "rightness," then we're back to (II) again.
It should of course be noted that this is just Socrates's position. Other philosophers would later go on to dispute the argument that (II) and (III) are incompatible, and/or that they are the only valid interpretations of (I). Aquinas in particular had a rather elaborate response to the dilemma (in the context of Christianity, so "all of the gods" is replaced with the monotheistic "God," but it's the same argument in either case). Depending on your views of religion, you may or may not find Aquinas's argument reasonable (in short, he argues that "good" and "God's will" are synonymous rather than one causing the other).