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Bisimilarities over stable configuration structures can be divided into three families. In the first one - including interleaving, step, pomset, and forward-reverse bisimilarities - no isomorphism is required between the events matched during the bisimulation game. In the second one - including weak history-preserving, weak history-preserving pomset, and weak hereditary history-preserving bisimilarities - a labeling- and causality-preserving isomorphism is required between matched events, which is specific to each pair of configurations related by the bisimulation relation and hence can vary for a matched event from pair to pair. In the third one - including history-preserving and hereditary history-preserving bisimilarities - a single isomorphism is built incrementally, which is therefore fixed for all matched events. We revisit true concurrency bisimilarities by introducing variants that additionally check that the backward ready multisets of related configurations coincide. While the distinguishing power of the bisimilarities of the second and third families does not change, the power of the revised bisimilarities of the first family is equal to that of the bisimilarities of the second family. The latter bisimilarities can thus be characterized by replacing variable isomorphisms with simply counting incoming transitions. In contrast, backward ready multisets are not enough to characterize the third family in the simultaneous presence of autoconcurrency and non-local conflicts. We show that a further check for the existence of diamond and half-diamond substructures is necessary in that case to achieve the same distinguishing power as incremental isomorphisms.