  We extensively explore three different aspects of Born-Infeld (BI) type
nonlinear $U(1)$ gauge-invariant modifications of Maxwell's classical
electrodynamics (also known as BI-type nonlinear electrodynamics) and bring
some new perspectives on these theories. First, within the framework of
exponential $U(1)$ gauge theory, it is explicitly proved that although the
electric field at the location of the elementary point charges is not finite,
but the total electrostatic field energy is finite. Motivated by this
observation together with a wealth of evidence, we conjecture that all theories
in 4-dimensional spacetime that belong to the BI family result in finite
self-energy for elementary charged particles. In higher dimensions, it is found
that the weak-field coupling limit of BI-type theories, which is identified as
the weak field limit of effective Euler-Heisenberg (EH) theory, does not
possess a regularizing ability to make the self-energy of a point charge
finite, which implies that the conjecture may not hold for some BI-type
theories. However, we explicitly prove that BI, logarithmic and exponential
$U(1)$ gauge theories in arbitrary dimensions result in finite self-energy as
well. Next, we classically study the problem of vacuum polarization effects and
then systematically make a connection between all BI-type theories and QED. It
is shown that all the BI-type theories classically predict the vacuum
polarization effects, in which the final results are exactly in one-to-one
correspondence with QED and the effective EH theory up to the leading order of
corrections. Finally, we present a new, simple proof indicating that ...
