Lili actually cried herself to sleep, and Bell carried her out of the dungeon. She woke up at Bell's place but both Hestia and Bell were out, so she just left. After that, Bell returned finding her gone and went to look for her.

That's why they met again at the fountain and formally established a contract. No, I didn't read the source. That was all just me filling in the blanks.

I love Aiz's dress design. Wakiness. Lili also randomly raised an arm while waiting for her death just to display hers. The character designer and director of this show know what they are doing.