[In the actual Twilight Bella is absolutely shocked that Edward knows her name in spite of her telling everyone in school to call her Bella instead of Isabella. The following seems better to me.]
“My name is Edward Cullen,” he continued. “I didn’t have a chance to introduce myself last week. You must be Bella Swan.”
And for a moment all my fear and uncertainty disappeared. I smiled and said, "It's finally working," not so much to him as to the world itself.
He was naturally confused and asked, "What?"
"Telling people to call me 'Bella' instead of 'Isabella'. You're the first person I haven't had to correct. Usually it takes at least three tries."
That exchange didn't make it so I was suddenly at ease with him, but it did push my dread to the back of my mind and the lighter note it set carried me through my interactions with him that day.