My business is on the Big Island of Hawaii. So, my answer may not apply as everyone here tends to be casually dressed. Myself and the many therapists that I work with wear a range of shorts from board shorts to very nice shorts (almost formal/business style for shorts). The important thing is that the shorts are not obviously worn or old and fit properly. It is not hard to dress professionally and still stay cool in hot weather.
When I first started in fitness I favored rather flashy outfits for teaching. But this was the style in the 80s and 90s. I found that the longer I worked in fitness, and the more I studied and learned and saw myself as a fitness professional rather than just a group exercise leader, the simpler my clothing became. Particularly when I was working in fitness full time only one of the places I worked allowed shorts for the staff. Those were tailored khaki, and not too short, worn with a club polo. This was in DC, which can be pretty hot in the summer. These days I do not wear shorts to teach, even if I am teaching ‘hot’ yoga in a 90 plus degree studio. Though if I were doing outdoor cycling, or something where the unwritten dress code favors shorts that would make a difference.
Here is the thing…. When I go to work with a client or with students I want them to perceive me as an exercise professional, with the emphasis on professional. A long time ago in an office where I was working one of the executives told me something I have always remembered…. she said that no one knew what my salary was, or who in our group had seniority. Anyone who had worked with me before came to me because I was very good at my job, but when working with the public who do not know you first impressions matter.
As an independent trainer you get to make a lot of choices, especially if you work for yourself, or with a company that does not have a uniform or a dress code. Not just how much skin do you feel comfortable showing, but: Do you see yourself as a teacher? As a coach? As someone who works within the health industry? Clearly it would be silly for you to have a suit and a tie as you train, but how casual you are with what you choose to wear will affect how people interact with you, just as much as a firm handshake and eye contact and communication skills and appropriate preparation for your session.