  Context. Recent studies on Kappa distribution functions invoked in space
plasma applications have emphasized two alternative approaches which may assume
the temperature parameter either dependent or independent of the power-index
$\kappa$. Each of them can obtain justification in different scenarios
involving Kappa-distributed plasmas, but direct evidences supporting any of
these two alternatives with measurements from laboratory or natural plasmas are
not available yet. Aims. This paper aims to provide more facts on this
intriguing issue from direct fitting measurements of suprathermal electron
populations present in the solar wind, as well as from their destabilizing
effects predicted by these two alternating approaches. Methods. Two fitting
models are contrasted, namely, the global Kappa and the dual Maxwellian-Kappa
models, which are currently invoked in theory and observations. The
destabilizing effects of suprathermal electrons are characterized on the basis
of a kinetic approach which accounts for the microscopic details of the
velocity distribution. Results. In order to be relevant, the model is chosen to
accurately reproduce the observed distributions and this is achieved by a dual
Maxwellian-Kappa distribution function. A statistical survey indicates a
$\kappa$-dependent temperature of the suprathermal (halo) electrons for any
heliocentric distance. Only for this approach the instabilities driven by the
temperature anisotropy are found to be systematically stimulated by the
abundance of suprathermal populations, i.e., lowering the values of
$\kappa$-index.
