  Gamma-ray bursts (GRBs) exhibit a diversity of spectra. Several spectral
models (e.g., Band, cutoff power-law, and blackbody) and their hybrid versions
(e.g., Band+blackbody) have been widely used to fit the observed GRB spectra.
Here, we attempt to collect all the bursts detected by {\it Fermi}-GBM with
known redshifts from July 2008 to May 2022, motivated to (i) provide a
parameter catalog independent from the official \emph{Fermi}/GBM team and (ii)
achieve a ``clean" model-based GRB spectral-energy correlation analysis. A
nearly complete GRB sample was created, containing 153 such bursts (136 long
gamma-ray bursts and 17 short gamma-ray bursts). Using the sample and by
performing detailed spectral analysis and model comparisons, we investigate two
GRB spectral-energy correlations: the cosmological rest-frame peak energy
($E_{\rm p,z}$) of the $\nu F_\nu$ prompt emission spectrum correlated with (i)
the isotropic-bolometric-equivalent emission energy $E_{\gamma, \rm iso}$ (the
Amati relation), and (ii) the isotropic-bolometric-equivalent peak luminosity
$L_{\rm p, iso}$ (the Yonetoku relation). From a linear regression analysis, a
tight correlation between $E_{\rm p,z}$ and $E_{\gamma, \rm iso}$ (and
$L_{\gamma,\rm iso}$) is found for both the Band-like and CPL-like bursts. More
interestingly, the CPL-like bursts do not fall on the Band-like burst Amati and
Yonetoku correlations, suggesting distinct radiation processes, and pointing
towards the fact that these spectral-energy correlations are tightly reliant on
the model-wise properties.
