Passing along some notes from the expanded meeting and gift exchange that the Vatican pool picked up:
The pope asked Melania Trump, referring to POTUS: "What do you give him to eat, pizza?" She repeated "pizza."
One of the pope's attendants, a clergyman, gave Mrs. Trump a small object, maybe a rosary, as she was walking away (she seemed to have forgotten about it), and she turned back and asked the pope to bless it, which he did.
Part of the pope's gifts to POTUS were three documents he usually gives to heads of state: Amoris Laetitia, Evangelii Gaudium and Laudato Si' -- plus this years World Day of Peace message about nonviolence. The last is the one he told POTUS, "I signed it personally for you."
Before the full delegation and pool were brought into the pope's study, and after POTUS's one-on-one meeting with the pope, Mrs. Trump, Ivanka Trump and Jared Kushner joined POTUS to meet briefly with the pope. Then the rest of the delegation and reporters were let into the room.
After Mrs. Trump's comment to the pope about visiting a hospital, they each wished each other "good luck."