If piety is capable of different definitions depending on which God's love is sought, that is not a definition of piety in principle; rather it is the same thing as saying that the definition of piety is determined by which God provides the definition. Euthyphro's Third Definition of Piety

"Yes, I should say that what all the gods love is pious and holy, and the opposite which they all hate, impious."

While this last attempt by Euthyphro to define piety resolves the problem of different opinions of different Gods introduced by his second attempt, it raises two other fundamental problems that are the two main aspects or "horns" of Euthyphro's dilemma. First, if pious is merely...

Had the Gods loved murder, then murder would be pious. The point is made by Socrates in the following question: "It is loved because it is holy, not holy because it is loved?"
Second, if piety exists independently of the will of the Gods, then, the Gods have not established what piety is; instead, they are merely messengers of something that exists apart from them and without their influence at all. This is also a fundamental inconsistency with the belief that morality in human life is derived from the will of Gods.