I would go with the picture that shows professionalism. When looking for a job, the first impression means alot.
When I interviewed for my current job. I couldn't decide on whether I should wear makeup or not. Since my hair is in a natural state. I wondered if that would be an issue. I was more concerned about how the director would perceive me. I am not a professional interviewer. I know for sure in corporate your appearance is everything.
In looking at the two pictures in question, the picture on the right is definitely a better image to present on a professional website. Though it is fun to let your hair down every now and then to show people your fun, quirky side, a professional website is not the place for it. Many companies now serf the web looking at websites of potential employees because they want to know what type of individual they are hiring. Though some say this is unethical, others feel that if you put your private life on a social platform, everyone should have a right to view it.
I would definitely have reservations about having professional connections with someone presenting themselves as the way this friend presents themselves in the photo on the left. I am not sure what the gesture with the two fingers is supposed to s