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We revisit a recently proposed scheme [M.F. Ciappina et al 2019 Phys. Rev. A 99 043405] for accurate measurement of electromagnetic radiation intensities in a focus of high-power laser beams. The method is based on the observation of multiple sequential tunneling ionization of atoms and suggests the determination of the peak intensity value from the appearance in the focal area ions with a certain charge number. Here we study the sensitivity of the approach to two essential factors: the focal volume effect and the tunneling ionization model chosen to calculate ionization rates. We show, on the one hand, that the results appear almost model-independent for ionization of helium- and hydrogen-like ions and, on the other hand, that the focal averaging leaves the intensity-dependent features visible. Our findings are in favor to a practical implementation of the atomic diagnostics of extreme laser intensities.