Do we really need alternatives to the $ω_0ω_a$CDM parameterization after the DESI DR2?

We introduce a density-level pivot construction for the Chevallier-Polarski-Linder (CPL) parameterization by defining the normalized dark energy density $f_p\equiv f_{\rm DE}(a_p)$ and the equation of state $\omega_p\equiv \omega(a_p)$ at an optimized pivot scale factor $a_p$. This reparameterization leaves the underlying CPL cosmology unchanged and allows the two models to be compared using parameters with a direct physical interpretation at the epoch where the data are most sensitive. Accordingly, following the DESI DR2 results, we compare a newly proposed $f_a f_b$CDM parameterization, based on a second-order Taylor expansion of the normalized dark energy density, with the standard $w_0w_a$CDM model. In particular, we constrain the original and pivoted parameterizations using compressed cosmic microwave background (CMB), DESI DR2 baryon acoustic oscillations (BAO), cosmic chronometers (CC), and Pantheon+ Type Ia supernovae data, with the SH0ES prior imposed on $H_0$. We find that applying the same density-level pivot prescription to the $w_0w_a$CDM model substantially reduces the correlation between its dark energy parameters and provides tighter and more stable constraints. The statistical comparison shows that this model remains favored over the Taylor expansion of the dark energy density, even when both models are analyzed in the optimized parameter basis. Moreover, the pivoted CPL parameterization accurately reproduces the background evolution of quintessence models, providing a reliable phenomenological approximation to the underlying dark energy dynamics, better than the $f_af_b$CDM model. We conclude that changing the parameter basis improves the performance of the $w_0w_a$CDM model, which emerges as the most suitable framework to describe the dark energy sector within the class of models considered here.