  (Abridged) The origin of dSphs in the Local Group (LG) remains an enigma. The
tidal stirring model posits that late-type, rotationally-supported dwarfs
resembling present-day dwarf irregular (dIrr) galaxies can transform into dSphs
via interactions with Milky Way-sized hosts. Using collisionless N-body
simulations, we investigate for the first time how tidal stirring depends on
the dark matter (DM) density distribution in the central stellar region of the
progenitor disky dwarf. Specifically, we explore various asymptotic inner
slopes gamma of the dwarf DM density profiles (rho \propto r^{-gamma} as r ->
0). For a given orbit inside the primary, rotationally-supported dwarfs
embedded in DM halos with core-like density distributions (gamma = 0.2) and
mild density cusps (gamma = 0.6) demonstrate a substantially enhanced
likelihood and efficiency of transformation into dSphs compared to their
counterparts with steeper DM density profiles (gamma = 1). Such shallow DM
distributions are akin to those of observed dIrrs, highlighting tidal stirring
as a plausible model for the LG morphology-density relation. When gamma <1, a
single pericentric passage can induce dSph formation and disky dwarfs on
low-eccentricity or large-pericenter orbits are able to transform into dSphs;
these new results allow the tidal stirring model to explain the existence of
virtually all known dSphs across a wide range of distances from their hosts. A
subset of rotationally-supported dwarfs with gamma <1 are eventually disrupted
by the primary; those that survive as dSphs are generally on orbits that are
biased towards lower eccentricities and/or larger pericenters relative to those
of typical CDM satellites. The latter could explain the rather peculiar orbits
of several classic LG dSphs such as Fornax, Leo I, Tucana, and Cetus.
