Everyone is correct regarding insurance, but what I would add is that being in San Diego, California, I added the park that I do outdoor training at, as well as schools, studios and gyms that I have/had trained clients at.
The park for me is very important as sometimes people are on uneven grounds, benches or other things the the city has in their parks which can provide a hazard for your clients if you’re not aware of where you’re training them.
The parks are not aware that I’ve added them as a “rider” (it’s free to do so), but I believe in staying 100% covered…just in case. =D
Harris is correct in saying that your personal trainer insurance should cover you. You have to look at the insurance question, though, as a studio owner and will need to get the studio insurance.
The question of living in rented space should not matter to the fitness insurance but you need to make sure that the property owner is okay with this.