Many diagnosis approaches for discrete event systems are diagnoser-based. A diagnoser is a deterministic automaton that can be built directly from the underlying system model by performing e-reduction and determinization operations. The diagnoser-based approaches allow for analyzing diagnosability, but they also support online diagnosis in a straightforward way. The procedure for investigating diagnosability on the diagnoser consists in checking the existence of indeterminate cycles, and requires to verify for every F-uncertain cycle in the diagnoser whether there exists two corresponding cycles in the system model such that one is fault-free while the other is faulty. The present work aims to improve the efficiency of the diagnoser-based approaches by establishing a diagnoser variant that offers a convenient structure to help enhance the diagnosability analysis procedure. Namely, it consists in separating the normal states from each faulty state classes in each diagnoser node. Such a distinction serves to track the faulty and fault-free sequences in the diagnoser paths more efficiently. On the basis of various features offered by the diagnoser variant, we put forward simplified necessary and sufficient conditions for two diagnosability properties: (i) diagnosis of fault occurrences and (ii) detection of fault absence, i.e., non-fault detection. Such conditions are established for both cases of a single fault class and multiple fault classes, using the notion of indicating sequences associated with the F-uncertain cycles in the diagnoser.