The ancient world is a term that covers a lot of space and a long range of time. For some of those places and times we don't know a whole lot about sexuality. But we do know that importing our own sexual distinctions is problematic. So with that, giant caveat, I will move into some sweeping generalities.

Let's talk about the Greeks. Lesbos owes its association with lesbianism to Sappho, A Greek poetess of the 7th/6th century. Sappho wrote lyric poetry, which survives in scattered fragments. Her poems were like this:

He’s equal with the Gods, that man

Who sits across from you,

Face to face, close enough, to sip

Your voice’s sweetness,

And what excites my mind,

Your laughter, glittering. So,

When I see you, for a moment,

My voice goes,

My tongue freezes. Fire,

Delicate fire, in the flesh.

Blind, stunned, the sound

Of thunder, in my ears.

Shivering with sweat, cold

Tremors over the skin,

I turn the colour of dead grass,

And I’m an inch from dying.

Now you might see where the idea that Sappho was a lesbian comes in. This poem is clearly a love poem and it seems to be written to a woman... right?

The thing is, this type of poetry belongs to a rich tradition of choral and epithalamic (wedding) songs which feature female speakers extolling the beauty of one of their companions. It's a literary trope and we have plenty of evidence for it. For example, Sappho's contemporary Alcman writes lyric poetry sung by choruses of girls where one girl is singled out and described by the others. This seems to have been a rather standard part of courting rituals at the time. (Maybe because sexual segregation in Greek communities meant that the men had to take the girl's peers praise as a way to decide on the most desirable wife.) What is so interesting about Sappho is that she is one of our only direct female voices to survive from ancient Greece.

The Greeks didn't seem to make much of the lesbian aspect, really. The biographical tradition of Sappho definitely presents her as sexual, but in fact her sexuality is targeted towards men. In a story that is definitely true she pissed off Aphrodite and was punished by falling in love with the ferryman, Phaon, who Aphrodite had magically rejuvenated. Sappho threw herself to her death when he rejected her.

The active/passive distinction that you make is one that's been argued for especially in the case of Roman sexuality. I would say that Greek is a bit like that, but also has an age component to sexuality. Amongst males, adolescent boys were the objects of affection for mature companions. There's a lot of regional/temporal variety and debate about the exact nature of these relationships and they were complicated by issues of social hierarchy. In Athens, for example, it seems that young noble boys were considered the ultimate object of desire. However, it wasn't right to penetrate a freeborn citizen. As these boys grew up it was assumed that they would transition to the active penetrative role and that they would then have sex with women and boys and even men who were foreigners/slaves and therefore had no dignity to preserve by their society's standards.

The choral poetry seems to suggest that a certain amount of homoeroticism between girls was acceptable or even desirable. To the men of Greek society, lesbianism basically didn't count. Men and women lived and slept separately and while Greek men essentially had to divorce their wives if they found them with another man, being with a woman simply didn't seem to strike their radar as a problem because it didn't threaten the legitimacy of their family.

Basically, we have almost nothing attesting to how women felt or characterized their sexuality in the Greek world and men didn't seem to perceive lesbianism as a "thing."

The idea of Sappho as a "lesbian" in our sense seems to come from Ovid. As a quick tl;dr on the Romans, they acknowledged lesbianism more than the Greeks did. Some, like Ovid, describe women so crazed by sex that they make no gender distinctions. Other, like Martial, think lesbianism is the worst (aside from maybe a man performing oral sex on a woman).