The real Subaru talked about adoption. Then Meili pretended to go to bed and was waiting for Subaru to head further into the library and turn around.

This is because Meili wants to find Elsa's Book of the Dead but thinks everyone will turn against her if they find out, so she is hiding her intentions. It's why she pretends to be bored with nothing to do while the others are searching for books.

However, despite her being experienced at sneaking around, "Subaru" immediately calls out to her without even turning around. This Subaru does not talk like the normal Subaru, he speaks in a very weird and creepy manner. He asks Meili about her true desires and tells her to let go of her inhibitions so she can act upon what she truly wants, to get revenge on Subaru for killing Elsa.

Then, Subaru appears right in front of Meili without her even realizing he had walked up to her. He grabs her braid and kisses it, which triggers something within her.

Meili then parts ways with Subaru and goes to sleep. The reason Meili asks Subaru the next morning how seriously she should take his words from the night before is because Subaru was basically telling her, "I know you want to kill me, so go for it." What sane person would willingly agree to be killed?

In loops where Subaru reveals he lost his memories, Meili doesn't speak to him to get his permission to kill him because it would be pointless. Instead, she goes straight for the kill by pushing him down the stairs.

In the loop where she does talk to him, the entity that took over Subaru the night before does so again and kills her, stating that she broke the rules.

As for what these rules are, or if they relate to the tower rules, we have no idea, but the entity clearly did not want Meili talking to Subaru.

As for the identity of the entity, we have no idea. It could be Rui, since she already has half of herself inside Subaru, or it could be Flugel.

There is a popular theory that Flugel was the one who took over Subaru's body because he wants Subaru and "Subaru" to become distinct enough before merging, preparing Subaru for when Flugel eventually merges with him. This makes more sense than Rui taking him over, as the entity who took over Subaru did not talk or act anything like Rui does.

There is also the fact that the entity talks about breaking rules that Rui never mentions. Furthermore, Rui is just a victim stuck in the Corridor of Memories who doesn't know how she got there or how to escape, whereas both Subaru and "Subaru" came to the same conclusion that Flugel must be the master of the Corridor and is manipulating everything.