To answer the question: "...is there a name for this matrix..."

Beyond the "Gaussian-Toeplitz matrix" mentioned in the comments, the said matrix is a special case of the *Gaussian Kernel* (which is also the alluded to "heat kernel"):

\begin{equation*}
k(x_i,x_j) = e^{-c(x_i-x_j)^2}\qquad x_1,\ldots,x_n \in \mathbb{R}.
\end{equation*}
Clearly, the matrix $A = [k(x_i,x_j)]$ is semidefinite. If the $x_i$s are unique, then it is strictly positive definite (this requires some more work to prove).

The Gaussian kernel is of course the canonical example of a *translation invariant kernel*, so searching more for such kernel matrices should bring up more examples, namely, matrices of the form $A_{ij} = k(x_i,x_j)=\varphi(x_i-x_j)$, for suitable $\varphi$.

Incidentally, in the paper mentioned by @Lucia, the smallest eigenvalue for the Gaussian-Toeplitz $A$ is bounded from below by (assuming $A$ is $n\times n$):
\begin{equation*}
\lambda_n \ge \frac{\det(A)}{n!}.
\end{equation*}
It seems possible to improve this to $\lambda_n \ge \frac{\det(A)}{2^{n-1}}$ by using the technique mentioned in my answer here.

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