These are not definitions of reversible and irreversible processes, but it is a description of one of the differences between the processes.
In real world situations, however, for some processes, the system undergoing the changes responds much faster than the applied change, the deviation from reversibility may be negligible. In a reversible cycle, the system and its surroundings will be exactly the same after each cycle. (see here)
A (...) definition of a reversible process is a process that, after it has taken place, can be reversed and causes no change in either the system or its surroundings. In thermodynamic terms, a process "taking place" would refer to its transition from its initial state to its final state. again, wikipedia