As I explained on page 25 of my Primer in Religion and Morality I think there are multiple dilemmas floating around under the name “Euthyphro Dilemma” (hereafter, ED).
ED: The literal, original formulation of the ED is this: “Is the pious loved by the gods because it is pious, or is it pious because it loved by the gods?”
In that formulation, it’s argubaly not really applicable to any contemporary discussions of theistic metaethics.
In order to make it applicable, people often revise it. For example, ED is most often presented in a revised version so as to be applicable to Divine Command Theories of right and wrong (DCT-D):
ED-D: “Is something morally obligatory because God commands it, or does God command it because it is obligatory?”
Theistic philosophers are well aware of ED and ED-D, and so they have formulated sophisticated responses.
First, they make a distinction between moral axiology (moral goodness and badness) and moral deontology (moral right and wrong). This allows theists (who are so inclined) to take different horns of the dilemmas posed by ED-D. For example, some theists say that moral axiology is independent of God, but moral deontology is dependent upon God.
Second, in the last fifty years theistic philosophers have formulated metaethical theories to avoid those dilemmas. Regarding moral value, some theists defend what I call the Divine Nature Theory (DNT-A): axiological properties are metaphysically grounded in God’s nature (or character).
Regarding moral obligation, some theists defend a Modified Divine Command Theory (MDCT-D): deontological properties are metaphysically grounded in the relevant commands of a loving God. MDCT-D successfully avoids ED-D, as defined above.
Of course, just as theistic metaethicists have formulated sophisticated metaethical theories in response to ED and ED-D, critics can and have formulated revised versions of their dilemmas.
In response to DNT-A, we get the version I call ED-A:
ED-A: Is God’s nature good simply because it is God’s nature, or is there some independent standard to which God’s nature conforms?
In response to MDCT-D, we get the version I call MED-D:
MED-D: Is something morally obligatory because a loving God commands it, or does God command it because it is the loving thing to do?
(See pages 20-25 of my Primer.)
As an aside, if I were a theist, I might subscribe to Adams’ Modified Divine Command Theory of right and wrong (MDCT-D), but I am certain that I would consider moral axiology (value) to be completely independent of God and His nature. Although I’ve read most of the secondary literature in the last 40-50 years on theistic metaethics, I’ve always considered it odd that some (not all) theists are not satisfied with
D: Moral obligation is somehow dependent upon God
but also want:
A: Moral value is somehow dependent upon God.
For those theists who want to affirm both D and A, I’m assuming the motivation is the desire to preserve divine aseity. I can understand that motivation, but I think it is misguided.