The answer to your question would be that the sentence that uses pronouns and antecedents correctly is the following one: We both signed up, but David took himself out of the tournament.
Reflexive pronouns end in -self and -selves that are used when the subject and the object of a sentence are the same. They can act as either objects or indirect objects. In this case, the reflexive pronoun functions as the direct object because in the sentence the subject and the direct object are the same. The subject is the NP "David", which is a proper name referring to a third person singular personal pronoun (he). Thus, the reflexive pronoun referring back to "David" should be "himself".