I do think that the white cane is an extension of the person as well. While it is polite for someone using a cane to modify their use of it in certain conditions - such as 'drawing in' the cane to create a shorter range and thus maneuver in a crowd - the person is human. Rose was, by OP's admission, already in the process of modifying her behavior when the incident happened, shortening her range as the OP moved closer to help guide her through the fresh rush of people. The child's mother should have been holding the child's hand, and for whatever reason was not. The child ran out, tripped over Rose's cane because they were not paying proper attention, as children often do. It was truly an accident. That mother would have screamed at Rose if she was simply standing still with the cane held flat against her leg and the child ran into her.
That last bit happened to me. I was standing in line at a theme park, not moving, and a child about age 4 ran into my leg at speed, bounced off, and began to cry. The parent rushed up and began to berate me. The people behind and in front of me were all staring at her like she was nuts, because I hadn't even done anything wrong. That parent left screaming insults at me all without acknowledging that they should have been watching their child better. The boy was dangling off the parent's arms, wailing and screaming. Not five minutes later, ANOTHER child ran into me. The guy behind me was cracking up, nearly crying because he was laughing so hard. This time, the parent was right behind, and apologized to me, saying the girl had yanked her hand out of her mother's and taken off running because she was mad that she couldn't go on a ride again. The girl was sniffling a little when the mother turned to her and said, "That's why you don't run away here. You might run into someone, or get lost. Are you hurt?" The girl checked, not even scuffed anywhere. "Well then, apologize to the nice woman for bumping into her." I accepted the apology, then told her, "Please be more careful, I wouldn't want you to get hurt." We wished each other a good day, and the pair left, child's hand firmly in her mother's, no tears anywhere. It really is all about the parenting.