Here's my theory on Lily. Though it's probably more like my personal "wishlist" of things wrong with her.
She's a closet pervert, one who's been trying to groom the MC. But at the same time, she's trying to resist her own urges. She doesn't like being slutty and despises slutty women. Since perverts can recognize other perverts, Lily knows what the other women on the island want. Young meat. Therefore, she doesn't just want to have the MC all to herself, she wants him to stay pure. That's her desire as his former caretaker and also because that's what turns her on the most. So when she sees him being horny or asking for sex during the day, it's annoying to her. Like she's not in the mood and doesn't want to be. Meanwhile, him having sexual experience is disappointing but also makes things easier. I think this is most evident when you finally do have sex with her. She's showing more frustration than anything. Telling you to hurry up and get it over with. To fuck her hard, because she might as well get off too. And if she cums before you, she's not going to finish you off. To me, these are signs of a woman who's either done sex work or was subjected to sexual abuse. Not as a child (necessarily), but as an indentured servant. Her former employer(s) must have fucked her several times. Enough for her to get used to it. That's why she likes it rough and how she can be so emotionally detached. Her body wants one thing while her mind wants another.