Translation:Zeus was the ruler of the ancient Greek gods.
Wrong term. I don't think that the Ancient Greeks ever used this term "αρχηγός", a more or less military term, which extended to include political power too. He considered as a father mainly of gods and men, a more paternal personality, but not only, https://greekgodsandgoddesses.net/gods/zeus/. But not αρχηγός, the leader of a military or political formation. I think so.
The term αρχηγός is correct. Remember the goddess Athena, Zeus' daughter, was born in full military armor with a spear. Chrysaor was born in full military armor too. Άρης was god of war. Apollo fought the Greek army in the Iliad. So the gods were involved militarily. They were soldier gods with wisdom and foibles. Soldiers require a general. Zeus was their leader. And he gave orders and punishment. And he was more. A husband and father to some of the gods.
Ζευς is the original Ancient Greek name. (It's just one syllable, so there would be no accent in the monotonic system. In polytonic, it's Ζεύς.)
The genitive of Ζευς was Διός and the accusative Δία.
As with many old third-declension nouns, the nominative singular was re-shaped from the old accusative, making it a first-declension noun: much as ο πατήρ - του πατρός - τον πατέρα became ο πατέρας - του πατέρα - τον πατέρα, so also Zeus became ο Δίας - του Δία - τον Δία, similar to the ancient pattern of ο νεανίας - του νεανίου - τον νεανίαν.
So there's no difference in meaning -- they're simply an Ancient/Modern pair.
Why not Ελληνικών ? This surely means "the gods of the ancient Greeks". Though I suppose the meaning is more or less the same!