  We apply the "systematic" $1^{st}$ order cosmological perturbation theory
method to re-derive the formulation of an inflationary model generated by
variation of constants, then to study the case where it is non-minimally
coupled to gravity within both the "Metric" and "Palatini" formulations.
Accommodating Planck 2018 data with a length scale $\ell$ larger than Planck
Length $L_{pl}$ requires amending the model. First, we assume $f(R)$ gravity
where we show that an $R^2$-term within Palatini formulation is able to make
the model viable. All along the discussions, we elucidate the origin of the
difference between the "Metric" and "Palatini" formalisms, and also highlight
the terms dropped when applying the shortcut "potential formulae method",
unlike the "systematic' method", for the observable parameters. Second, another
variant of the model, represented by a two-exponentials potential, fits also
the data with $\ell> L_{pl}$.
