  We reanalyze the prompt emission of two of the brightest Fermi GRBs (080916C
and 090926A) with a new model composed of 3 components: (i) a thermal-like
component--approximated with a black body (BB)--interpreted as the jet
photosphere emission of a magnetized relativistic outflow, (ii) a non-thermal
component--approximated with a Band function--interpreted as synchrotron
radiation in an optically thin region above the photosphere either from
internal shocks or magnetic field dissipation, and (iii) an extra power law
(PL) extending from low to high energies likely of inverse Compton origin, even
though it remains challenging. Through fine-time spectroscopy down to the 100
ms time scale, we follow the smooth evolution of the various components. From
this analysis the Band function is globally the most intense component,
although the additional PL can overpower the others in sharp time structures.
The Band function and the BB component are the most intense at early times and
globally fade across the burst duration. The additional PL is the most intense
component at late time and may be correlated with the extended high-energy
emission observed thousands of seconds after the burst with the Fermi/Large
Area Telescope (LAT). Unexpectedly, this analysis also shows that the
additional PL may be present from the very beginning of the burst. We
investigate the effect of the three components on the new time-resolved
luminosity-hardness relation in both the observer and rest frames and show that
a strong correlation exists between the flux of the non-thermal component and
its E$_{peak}$ only when the three components are fitted simultaneously to the
data (i.e., F$_i^{NT}$-E$_{peak,i}^{NT}$ relation). In addition, this result
points toward a universal relation between those two quantities for all GRBs
when transposed to the central engine rest frame (i.e.,
L$_i^{NT}$-E$_{peak,i}^{rest,NT}$ relation).
