The first is a natural light photographer. Although there are a few photos that are very well-lit, many of them encounter the same problems you'll always see with natural light and open shade- flat faces at best, and pronounced dark circles under the eyes and gray, sallow skin tones at worst. Depending on the weather that day, you're either going to have usable headshots, or bitter disappointment.
But between these two options, you really only have one choice. The second photographer is strictly a family/baby photographer. Her work is totally unacceptable for use as working headshots.