The 17th Century philosopher John Locke described a case like the one you considered, and he too thought that Barry and Gary had swapped bodies. What was important for Locke was that, after the procedure, the Gary-body person had Barry’s memories. Philosophers who follow Locke in thinking that it is continuation of psychological states, such as memories and beliefs, which are important to our survival, are called Lockeans.
Some philosophers have thought that psychological states are not what determine our existence, however. For instance, it both seems that we were once foetuses in our mother’s wombs, and also that such foetuses (under 5 months) don’t have any psychological states. But if both of these claims are true, then we can exist without any psychological states at all.