The girl wearing a brown jacket whilst walking in snow.
The lady wearing a jacket is looking for her lost dog.
A girl can be very young and as such referring to her as a lady may not be the most suitable title. A girl can walk in snow for reasons other than looking for her lost dog.
Just because the girl is walking in snow doesn't mean the lady is looking for her lost dog.
Looking for a lost dog is not the same as walking in the snow.