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Recently it has been proved that a simple algorithm configurator called ParamRLS can efficiently identify the optimal neighbourhood size to be used by stochastic local search to optimise two standard benchmark problem classes. In this paper we analyse the performance of algorithm configurators for tuning the more sophisticated global mutation operator used in standard evolutionary algorithms, which flips each of the $n$ bits independently with probability $χ/n$ and the best value for $χ$ has to be identified. We compare the performance of configurators when the best-found fitness values within the cutoff time $κ$ are used to compare configurations against the actual optimisation time for two standard benchmark problem classes, Ridge and LeadingOnes. We rigorously prove that all algorithm configurators that use optimisation time as performance metric require cutoff times that are at least as large as the expected optimisation time to identify the optimal configuration. Matters are considerably different if the fitness metric is used. To show this we prove that the simple ParamRLS-F configurator can identify the optimal mutation rates even when using cutoff times that are considerably smaller than the expected optimisation time of the best parameter value for both problem classes.