I have to disagree with most of the current answer, in that "Yes, we can rule it out". I do agree that the show shot the scene for dramatic effect. However...
In the chapter that Bran awkens he is not warged into Summer. He was with the three-eyed raven, who was telling Bran he must fly. He only see the part of the interaction down on the Kingsroad
He saw his father pleading with the king, his face etched with grief. He saw Sansa crying herself to sleep at night, and he saw Arya watching in silence and holding her secrets hard in her heart. - A Game of Thrones, Chapter 16
However, right after that he then has a vision of the Free Cities, Vaes Dothrak, and Asshai.
Finally Bran finds himself seeing North of the Wall into the Lands of Always Winter. Bran is falling and the three-eyed raven is trying to get Bran to fly.
North and north and north he looked comedy the curtain of light at the end of the world, and then beyond that curtain. He looked deep into the heart of Winter, and then he cried out, afraid, and the heat of his tears burned on his cheeks.
'Now you know', the crow whispered as it sat on his shoulder. Now you know why you must live.
'Why?' Bran said, not understanding, falling, falling. Because winter is coming
Bran figures out how to fly and it seems the three-eyed raven opens Bran's third eye as well.
It took to the air, flapping its wings in his face, slowing him, blinding him. He faltered in the air as its pinions beat against his cheeks. It's being stabbed at him fiercely, and Bran felt a sudden blinding pain in the middle of his forehead, between his eyes.
'What are you doing?' He shrieked.
the crow opened its beak and cawed at him, a shrill scream of fear, and the grey mist shuttered and swirled around him and ripped away like a veil, and he saw that the crow was really a woman, a serving woman with long black hair, and he knew her from somewhere, from Winterfell, yes, that was it, he remembered her now, and then he realized that he was in Winterfell, in a bed and some chilly tower room, and the black haired woman dropped a basin of water to shatter on the floor and ran down the steps shouting, 'he's awake, he's awake, he's awake'.
So we see that it is the three-eyed raven that woke Bran. I do not feel that it could have been the killing of Lady because Bran has not even show a connection to Summer at this point.
Again, the show was going for dramatic effect, but I feel that since the show was being very faithful the source in Season 1, that an answer from the book is appropriate.