One has the ordering of the three $\lambda$'s, i.e.
$$
\lambda_{\text{convex}} \leq \lambda_{\text{LSI}} \leq \lambda_{\text{SG}},
$$
where $\lambda_{\text{convex}}$ is the one from convexity, $\lambda_{\text{LSI}}$ is the one from the inverse Log-Sobolev constant (both as in your question) and $\lambda_{\text{SG}}$ the spectral gap, characterized via the smallest constant in the weighted Poincaré inequality
$$
\Vert f -1 \Vert_{L^2(\pi)}^2 \leq (2\lambda)^{-1} \int |\nabla f |^2 d\pi , \qquad \forall f : \int f \, d\pi = 1.
$$
here $f$ plays the role of $d\rho/ d\pi$. The first inequality is the HWI-inequality after Otto-Villani and the second is linearization of the LSI.
The equivalence $\lambda_{\text{convex}} = \lambda_{\text{LSI}} = \lambda_{\text{SG}}$ holds for $V$ being a positive quadratic form, i.e. $V(x) = x \cdot H x$ for a fixed symmetric positive matrix $H$. Then, the eigenvalues and eigenvectors of the according Ornstein-Uhlenbeck process are explicit (products of Hermite-polynomials) and obtained in dependence of the eigenvalues of $H$. The smallest non-negative eigenvalue is then the smallest eigenvalue of $H$, in accordance to the $\lambda$-convexity. The LSI is sandwiched inbetween anyways.
Except for this particular case, I expect that the equivalence breaks down for generic $V$ (non-quadratic).
This is easiest observed in the non-convex case. So let, $V$ be a double-well, with local max in $0$ and two minima in $~\pm 1$ and have convex quadratic growth outside a bounded region. Then, the Fokker-Planck evolution is $\lambda$-convex for a $\lambda<0$ being the lowest bound on the Hessian again, determined by the non-convexity of $V$ around the local maximum. However, the Log-Sobolev and spectral gap constants are still positive finite and can be obtained by combining the Bakry-Emery criterion with the Holley-Stroock perturbation principle, since $V$ is a bounded perturbation of a convex potential.
This becomes even more apparent, by considering the vanishing diffusion limit, i.e. consider for $\varepsilon>0$ the Fokker-Planck equation
$$
\partial_t \rho = \varepsilon \Delta \rho + \nabla \cdot(\rho \nabla V)
$$
In this case, it becomes clear that the time-scales captured by convexity and Log-Sobolev constants or spectral gaps are rather different.
From the comment of @Tobsn follows that, convexity measures local stability at every point in the space of probability measures and the setting of double-well potential it is readily checked that, one actually gets the opposite comparison of the type
$$
W(\rho_t, \hat\rho_t) \geq c e^{|\lambda| t} W(\delta_{-\eta},\delta_{\eta}) \quad\text{for } t \in [0, t_0]
$$
where $\eta>0$ is small and $t_0$ is also not too large. I write $|\lambda|=-\lambda$ to make clear, that the trajectories expand and do not converge.
This follows, because $\rho$ and $\hat\rho$ will follow mainly the deterministic ODE $\dot X_t = - \nabla V(X_t)$, which expands at rate $|\lambda|$ close to the local maximum. In particular, this result also holds for $\varepsilon=0$.
In comparison, the log-Sobolev constant and the inverse spectral gap measure a global time-scale quantifying the ergodicity in time, showing that the diffusion converges to the measure $\pi$ in entropy or the relative density $f_t = \rho_t / \pi$ in the $L^2(\pi)$-sense. Those are global averaged quantities and hence behave in general better, as the non-convex setting shows. However, note that for the $\varepsilon$-dependent case, both $\lambda_{LSI}$ and $\lambda_{SG}$ will degenerate to $0$ exponentially like $e^{-C/\varepsilon}$, which is a statement about metastability being present in this setting (the limiting ODE dynamic is non-ergodic, since it has actually three stationary states).
There is a theory of variable Ricci bounds, which can catch a bit this different local stability and might improve the gap between those constants. One, can probably think of the Bracamp-Lieb inequality as an instance of variable curvature, since it shows that $\text{Hess}\, V$ behaves like the local Ricci-tensor for the diffusion, that is
$$
\Vert f - 1 \Vert_\pi^2 \leq \int \left\langle \nabla f , \text{Hess} V\, \nabla f \right\rangle d\pi .
$$
Other ways, to make the different behaviour of the constants precise, at least in the scaling regime with vanishing diffusion $\varepsilon\ll 1$, are in 1d upper and lower estimates on the Log-Sobolev constant and spectral gap with the help of the Bobkov-Götze or Muckenhoupt criterion.
I can provide more details or references on the individual mentioned observations, but I'm not aware of a general result showing that for generic V (not perfectly quadratic potentials), there are strict inequalities between all of the three $\lambda$'s, i.e.
$$
\lambda_{\text{convex}} < \lambda_{\text{LSI}} < \lambda_{\text{SG}}.
$$
Best Answer
Probably you mean Gross's Log-Sobolev Inequality? I cannot help with understanding any variational calculus proofs, but there are other proofs I find easier to understand.
Actually, I prefer obtaining the Hypercontractive Inequality first, $\|P_t f\|_q \leq \|g\|_p$ provided $e^{-2t} \leq \frac{p-1}{q-1}$, and then recovering Log-Sobolev by taking $q = 2$, squaring both sides, and differentiating at $t = 0$. (See, e.g., exercise 10.23 at http://analysisofbooleanfunctions.org ) As for the Hypercontractive Inequality itself, the most traditional short way to prove it (due to Gross, but also independently to Bonami previously) is to prove it first for Boolean functions (by induction on $n$) and then to pass to the Gaussian setting using the Central Limit Theorem. Janson's book Gaussian Hilbert Spaces (Chapter 5) does this nicely (or you can see Chapters 9, 10, 11 at the aforementioned web site). One can also prove Log-Sobolev directly by induction in the Boolean case (see, e.g., Exercise 10.26).
If one only cares about the Gaussian setting, there are other short routes to Hypercontractivity (and hence Log-Sobolev). There is the "Neveu method"; although it's short, it uses stochastic calculus and is thus not very elementary. On the other hand, a very direct and simple proof is given in Ledoux's recent paper "Remarks on Gaussian Noise Sensitivity...", based on early ideas by Hu and by Mossel--Neeman.