You, it seems, are sticking with animalism.
Just as destroying a table, but having an exact replacement, can be as good as having the original, but without actually having the original, perhaps teletransportation gives us (almost) everything we want without us actually surviving.
Although not an animalist himself, Derek Parfit has argued that it is not our continued existence that matters to us, but rather that there is some being which has our psychological states. If this is correct, then perhaps this can explain why some people have the inclination to think that Barry and Gary had switched bodies, even though that’s not literally the case.
To learn more about this topic, you might like to read Chapter 1 of Harold Noonan’s book Personal Identity, or the Stanford Encyclopaedia of Philosophy entry on personal identity (http://plato.stanford.edu/entries/identity-personal/).
At Southampton, the question of personal identity can be studied as part of the second-year Metaphysics module. In recent years, a number of our students have also written their third-year dissertations on personal identity. Other modules that consider aspects of our nature and self-conception include Origins of Existentialism, Philosophy of Mind, Self-Knowledge, and Freedom and Responsibility.
Southampton philosophy lecturers interested in personal identity and related topics include Dr. Elselijn Kingma, Dr. Conor McHugh, and Dr. Lee Walters.