  The simplest electromagnetic fields' (general- as well as
special-relativistic) classification is formulated which is based on physically
motivated ideas. According to this classification these fields can belong to
three types (electric, magnetic and null), each of them being split in pure and
impure subtypes. Only pure null type field propagates with the fundamental
velocity $c$, all other fields have the propagation velocity less than that of
light. The reference-frame-based methods of elimination of alternative
three-fields (e.g., magnetic in the electric type case) are given for pure
subtypes; for pure null type the generalized Doppler effect takes place
instead. All three types of impure fields are shown to be {\bf E}-{\bf
B}-parallelizable. Thus such an elimination in pure non-null and
parallelization in all impure cases mean transformation to the reference frame
co-moving with the electromagnetic field in which the Poynting vector vanishes.
The methods we propose modernizing the Rainich--Misner--Wheeler approach, also
permit to construct new exact Einstein--Maxwell solutions from already known
seed solutions. As examples, the Kerr--Newman and Li\'enard--Wiechert solutions
are considered, three ``new'' types of rotating charged black holes (with the
same Kerr-Newman geometry) are presented, and new physical effects are
evaluated. PACS 2008 Numbers: 04.20-{\bf q}, 04.20.Ex, 04.40.Nr, 04.70.Bw
