Let us consider the following elliptic quasilinear problem with mixed boundary conditions $$ \left.\begin{alignedat}{2}-\textrm{div}[a(x,u,\nabla u)]+c(x,u,\nabla u) & =g & & \textrm{in}\,\Omega\\ \nu\cdot a(x,u,\nabla u)+b(x,u) & =h & & \textrm{on}\,\Gamma_{N}\\ \left.u\right|_{\Gamma_{D}} & =\psi & & \textrm{on}\,\Gamma_{D} \end{alignedat} \right\} $$ where $\Omega$ is a Lipschitz domain whose boundary is divided in two disjoint open subsets $\Gamma_{N}$ and $\Gamma_{D}$ and $\nu$ is the external normal to the boundary. The weak formulation of the problem gives
$$ \int_{\Omega}\biggl(\left\langle a(x,u,\nabla u),\nabla v\right\rangle +c(x,u,\nabla u)\,v\biggr)\,dx\,+\int_{\Gamma_{N}}b(x,u)\,v\,ds=\int_{\Omega}g\,v\,dx+\int_{\Gamma_{N}}h\,v\,ds. $$ The test functions $v$ belong to the vector space $W_{\Gamma_{N},0}^{1,p}(\Omega)$. Subsequently, in order to interpret the established integrals as duality pairings, it is essential for the Carathéodory functions $a,b$ and $c$ to meet appropriate growth conditions. In particular, for the first term $$ \int_{\Omega}\left\langle a(x,u,\nabla u),\nabla v\right\rangle \,dx $$ since $\nabla v\in L^{p}(\Omega,\mathbb{R}^{d})$ we want $a(x,u,\nabla u)\in L^{p'}(\Omega,\mathbb{R}^{d})$ where $p'$ is the conjugate exponent of $p$. Via the compact embedding $W^{1,p}(\Omega)\Subset L^{p^{*}-\varepsilon}(\Omega)$ (where $\varepsilon\in\left(0,p^{*}-1\right]$) and denoting by $W^{1,p}(\Omega)_{w}$ the Sobolev space $W^{1,p}(\Omega)$ equipped with the weak topology, we get the Nemytskii operator $$ N_{a}\in B\,\left(W^{1,p}(\Omega)_{w}\times L^{p}(\Omega,\mathbb{R}^{d}),L^{p'}(\Omega,\mathbb{R}^{d})\right) $$ provided that $$ \left\Vert a(x,u,\nabla u)\right\Vert _{\mathbb{R}^{d}}\leq\gamma_{a}(x)+C\left|u\right|^{\frac{p^{*}-\varepsilon}{p'}}+C\left\Vert \nabla u\right\Vert _{\mathbb{R}^{d}}^{\frac{p}{p'}}\qquad\textrm{with}\quad\gamma_{a}\in L^{p'}(\Omega). $$ By making a similar argument, I determine that for function $c$, the appropriate growth condition should be
\begin{equation}\label{eq:1} \left|c(x,u,\nabla u)\right|\leq\gamma_{c}(x)+C\left|u\right|^{\frac{p^{*}-\varepsilon}{p^{*'}}}+C\left\Vert \nabla u\right\Vert _{\mathbb{R}^{d}}^{\frac{p}{p^{*'}}}\qquad\textrm{with}\quad\gamma_{c}\in L^{p^{*'}}(\Omega)\qquad\qquad(1) \end{equation}
since $v\in W_{\Gamma_{N},0}^{1,p}(\Omega)\subseteq W^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega).$ Instead, In the book ``Nonlinear Partial Differential Equations with Application'' (second edition) by Roubicek, at page 46, it is stated
\begin{equation}\label{eq:2} \left|c(x,u,\nabla u)\right|\leq\gamma_{c}(x)+C\left|u\right|^{p^{*}-\varepsilon-1}+C\left\Vert \nabla u\right\Vert _{\mathbb{R}^{d}}^{\frac{p}{p^{*'}}}\qquad\textrm{with}\quad\gamma_{c}\in L^{p^{*'}}(\Omega).\qquad\qquad(2) \end{equation}
My explanation is the fact that since $$ \frac{p^{*}-\varepsilon}{p^{*'}}=p^{*}-\varepsilon-1+\frac{\varepsilon}{p^{*}}>p^{*}-\varepsilon-1 $$ whether $\left|u\right|\geq1$ if (2) is satisfied then it is also (1). Since our primary concern when addressing growth conditions lies in understanding the behavior at infinity, one plausible explanation for the author's choice to consider (2) could be its simplicity or convenience for certain reasons.
My final question is: Could you please explain the difference between the two growth conditions for function $c$?