An analysis based on the comparison between singularities of speckle phase and pseudophase in the practice of optical vortex metrology is carried out by measuring the phase map of the speckle pattern obtained from the transmitted light through binary diffusers. In the characterization of the core structure of both phase singularities, the variation of the measured parameters is produced by means of in-plane linear displacements and rotations of the scattered speckle fields. These fields are addressed by using similar displacements of the binary phase masks recorded in a spatial light modulator (SLM). We complete these comparisons by measuring out-of-plane variations of the core structure parameters. In addition, we verified that the phase map of the transmitted light beam through the binary diffusers recorded in SLMs is actually characterized by a speckle phase with vortices of unitary charge. The presented analysis would be helpful for understanding the scope and limitations of the use of the singularities of speckle phase and pseudophase as position marking, and also for speckle measurement of in-plane rigid-body displacements of binary diffusers dynamically controlled by means of SLMs.