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We experimentally investigate the synchronous response of a semiconductor laser to the injection of a periodic or chaotic oscillating optical signal that is generated by a similar semiconductor laser with optical feedback. We show that there are two different types of synchronous response, appearing in separate regimes of laser frequency detuning and injection strength. They are distinguished by the time lag of the slave-laser response with respect to the injection signal from the output of the master laser. The experimental observations are well described by a numerical model consisting of a set of rate equations. It is revealed that the first type of synchronous response corresponds to the complete synchronization solution of the equations and the second type of response is the result of strong driving. The relevance of these two types of synchronous behavior to a number of recent experiments on chaos synchronization and their implications for data encoding/recovery using chaotic carriers are discussed.