I don't know a particular name for these matrices but they are special cases of matrices which have been considered in lattice theory. More precisely, let $(L,\wedge)$ be a finite semilattice and $\phi:L\to \mathbb R$ any function (one can replace $\mathbb R$ by any commutative ring). Now consider the matrix $$A:=(\phi(x\wedge y))_{x,y\in L}.$$ Your matrices are obtained by choosing for $L=\{1,\ldots,n\}$ with its natural order and $f(i)=a_i$ because then $f(i\wedge j)=a_{\min(i,j)}=\min(a_i,a_j)$.

Lindström (Determinants on semilattices. Proc. Amer Math. Soc. 20 (1969), 207–208) and, independently Wilf (Hadamard determinants, Möbius functions, and the chromatic number of a graph. Bull. Amer. Math. Soc. 74 (1968), 960–964) computed the determinant as $$\det A =\prod_{x\in L}\left(\sum_{y\in L}\mu(y,x)\phi(y)\right)$$ where $\mu$ is the Möbius function on $L$. In your case this means $$ \det A=a_1(a_2-a_1)\ldots(a_n-a_{n-1}). $$

I don't know a particular name for these matrices but they are special cases of matrices which have been considered in lattice theory. More precisely, let $(L,\wedge)$ be a finite semilattice and $\phi:L\to \mathbb R$ any function (one can replace $\mathbb R$ by any commutative ring). Now consider the matrix $$A:=(\phi(x\wedge y))_{x,y\in L}.$$ Your matrices are obtained by choosing for $L=\{1,\ldots,n\}$ with its natural order and $f(i)=a_i$ because then $f(i\wedge j)=a_{\min(i,j)}=\min(a_i,a_j)$.

Lindström (Determinants on semilattices. Proc. Amer Math. Soc. 20 (1969), 207–208) and, independently Wilf (Hadamard determinants, Möbius functions, and the chromatic number of a graph. Bull. Amer. Math. Soc. 74 (1968), 960–964) computed the determinant as $$\det A =\prod_{x\in L}\left(\sum_{y\in L}\mu(y,x)\phi(y)\right)$$ where $\mu$ is the Möbius function on $L$. In your case this means $$ \det A=a_1(a_2-a_1)\ldots(a_n-a_{n-1}). $$