Hearing that he hadn't, in fact, been rude brought a brighter smile to the Doctor's face. Encouraged, even, to feel like he had already started to help in some fashion. She had been an anxious mess before, but to have a laugh was a start. Progress, no matter how slight, was still moving forward.
Once he received her name, he repeated it once as though testing the weight of it on his tongue and nodded. That suited her nicely. "Hm? No, no. Not 'a' doctor. Have been one, of course. Good field! Not so far from mine. I'm the Doctor. That's my name. Call me 'Doctor.'"
From the point of contact, his eyes darted about their surroundings. The cold didn't bother him, but he sensed the change. As quickly as she pulled back, he decided on a different route. Pushing her to see the silver lining could create the fragility present in such a devastating force like ice; he didn't want that. He wanted her to feel safe so that she could reflect on her plight with a clearer mind.
He understood how it felt to be fearful of one's own power.
Instead of appearing startled or upset, the Doctor's expression remained calm and filled with transparent understanding. And he was soon hearing the root of her worries: death. Glancing down at his feet as she took a seat, the Doctor's confidence wavered. He couldn't ever help himself when it came to his fear of death, how could he hope to help her? But, that was an awful way of looking at this; it wasn't about him. He was the Doctor and she needed help, nothing else needed to matter.
Crouching down in front of her, his expression looked nothing if not determined. "I can stand the cold, for starters," he offered. The subject of her death was set aside momentarily. "Whether from ice or a cold chill brought by the precipice of demise, I've weathered it all, Elsa. And look at me, I'm just a clumsy bloke that slides about on the ground because looking ahead was better than looking down. And it is. Don't say that you died, know that you lived and came back to do a great deal more of it, hm? Sounds much better."