Well, one of the problems with the Ancient Greek historical record is that it was created almost entirely by men within a very patriarchal culture. Women were expected to remain in their homes and female literacy was appallingly rare. Women were thought to have, by nature, less self-control than men and deeper sexual appetites, so they had to be carefully kept away from men from outside their families. What happened in the women's quarters and at any of the several all-female festivals is pretty much lost to us.

One of the most celebrated poets of antiquity was a woman named Sappho who wrote very emotional poems about her love for another woman. She was from the island city-state of Lesbos and was far and away its most famous daughter, so she was sometimes just called the Lesbian, as in "the woman from Lesbos." Eventually that word came to have another meaning...

Her love poetry was read extremely widely and was hugely influential, establishing a bunch of new meters and basically founding a whole genre of poetry called "lyric." When later Greek poets wrote emotional love poems of their own, they would often explicitly cite Sappho as the reason they were able to put their feelings in words... because her poems taught them how to speak and use the delicate language of love despite their otherwise very strictly manly-man masculinity.

I'm not aware of Ancient Greek commentators---though there may be some---who criticized Sappho for her sexual orientation. There may have been a few, but the silence of everyone else on the issue is, I think, much more informative.

Talking about homosexuality in Ancient Greece is difficult mostly because of the connotation modern society has about the word, love and relationships. Male homosexually is often discussed as pederasty. Where an older man (erastes) is the sexual and social mentor of a youth (eromenos). While affection and love, as modern conception understands it, are a part of this relationship it is not the dominating feature of the relationship. Pederasty was used as an introduction to civilized society for young men and it became heavily formalized in Cretan and Spartan societies, while remaining less formalized in Athenian society. Though much of what is known is understood from paintings on vases. Interestingly William Percy in his book 'Pederasty and Pedagogy in Ancient Greece' argues that most of the physical relationships between the lovers was intercrural to avoid the negative stigmatism of penetration.

Women's homosexually has much less evidence. Most of this is due to the social position of women in Classical Greece. Athens in the Classical Era thought that a woman's place was in the home. And women of the time, especially higher class women, were very much secluded. They were often uneducated, hidden, and not thought worthy of talking about by male authors of the time. There purpose was to maintain the home, be of value in marriage transactions, and produce children. There is very little that has survived about women's day to day practices. Therefore their sexual practices and possible homoerotic love might not be recorded but could possibly have been a part of their life and initiation into adulthood just as their male counterparts. But we do not have evidence for this. In Sparta women were given much more freedom and rights. They were allowed property and were respected in ways that Athenian women were not. However, as I have only studied Spartan society and women's roles sparsely I don't feel confident in giving accurate information about it.

There is however one very obvious exception to this: Sappho. The sixth century B.C. poet from Lesbos wrote elegantly and beautifully about physical and emotional relationships with women. She, however, should not be taken as an example of women's place in society. Highly educated and living in generation before the Classical Era, she was removed enough to not be constrained to the highly sheltered and dominated position of Classical Athenian women.

Therefore while there is poetry by Sappho about homosexual relationships between women, there is very little evidence of it. Not because it didn't happened but because surviving sources are generally male dominate and don't address women's relationships.

Sources: Percy, III, William A. Pederasty and Pedagogy in Archaic Greece

Women's Life in Greece and Rome in Translation, edited by Mary R. Lefkowitz and Maureen B. Fant

Homosexuality in Greece and Rome by Thomas K. Hubbard.

In Symposium by Plato, the character Aristophanes tells a story about the origin of man, wherein he states that humans initially had two heads, four arms, four legs etc like pairs of Siamese twins. The two halfs could be of same or different sex (ie. they were hermaphrodites). Because these "double"-humans were too powerful, the gods split them in halfs, which is how humans look now.

These separated humans search for and desire their lost half to regain a sense of completeness. The ones who came from a one-sex twin (whether male or female) would search for a "better half" of the same sex, while those from a two-sex pair would search for a partner of the opposite sex. He goes on to declare that the male-male love and desire is the most best and highest.

He doesn't offer any details on the female-female relations except to acknowledge the existence, but it is clear from the story that female-female love and desire is considered of basically the same nature as male-female and male-male. So this indicates that Plato openly acknowledges the existence of what we would call lesbians, even if he considers it quite uninteresting.

What were their conceptions of sexuality? Did they view sexual relationships between the same gender as homosexual or were there nuanced levels that were accepted, rejected, and so on?

Also why was this tradition not perpetuated through Western society or if it was, in what ways?