Regular black holes as an alternative to black bounce

The so-called black-bounce mechanism of singularity suppression, proposed by Simpson and Visser, consists of replacing the spherical radius $r$ in the metric tensor with $\sqrt{{r}^{2}+{a}^{2}}$, $a=\text{const}>0$. This removes a singularity at $r=0$ and its neighborhood from space-time and there emerges a regular minimum of the spherical radius that can be a wormhole throat or a regular bounce (if located inside a black hole). Instead, it is proposed here to make $r=0$ a regular center by proper (Bardeen type) replacements in the metric, preserving its form at large $r$. Such replacements are applied to a class of metrics satisfying the condition ${R}_{t}^{t}={R}_{r}^{r}$ for their Ricci tensor, in particular, to the Schwarzschild, Reissner-Nordstr\"om, and Einstein-Born-Infeld solutions. A simpler version of nonlinear electrodynamics (NED) is considered, for which a black hole solution is similar to the Einstein-Born-Infeld one but is simpler expressed analytically. All new regular metrics can be presented as solutions to NED-Einstein equations with radial magnetic fields.