Is there a simple way of proving that $\text{GL}_n(R) \not\cong \text{GL}_m(R)$?

If the ring is a field $F$ with characteristic not equal to $2$, then it is not hard to prove that the largest $k$ with $C_2^k \le {\rm GL}(n,F)$ is $k=n$.

We prove this by induction on $n$. Since $-1$ is the only element of order $2$ in $F$, the result holds for $n=1$, so assume that $n>1$. Then $k>1$, so we can choose an element $x$ in this subgroup with $x \ne -I_n$. Then $x$ is conjugate to a diagonal matrix with $1$s and $-1$s on the diagonal, and its centralizer is isomorphic to ${\rm GL}(m,F) \times {\rm GL}(n-m,F)$ for some $m$ with $1 \le m < n$. Now the result follows by induction.

I don't know whether there is an equally elementary argument for fields of characteristic $2$.

Let $G=GL_n(K)$ where $K$ is a field of characteristic not 2. Let $A$ be a maximal subgroup of $G$ of exponent 2. As every element of order 2 of $GL_n(K)$ is diagonalizable (with $1$ or $-1$ as eigen-values) and since $A$ is abelian, elements of $A$ are simultaneously diagonalizable. Hence we may assume that $A$ consists of matrices with 1's and $-1$'s on the diagonal. Therefore $A$ is isomorphic to the direct sum of $n$ copies of $\{1, -1\}$ and hence has order $2^{n}$. This shows that $GL_n(K)$ determines $n$. The same argument works for $SL$ instead of $GL$ if $n \geq 2$. (I just noticed that this same answer but with a different argument was given above.)

Here is a question: Assume characteristics are not 2. I can show in a quite elementary way that if the statement $SL_2(K) \simeq SL_2(L) \implies K \simeq L$ holds, then for $n \geq 2$, the statement $SL_n(K) \simeq SL_n(L) \implies K \simeq L$ holds. But I do not know how to prove this for $n=2$ in its full generality. We can of course assume the groups i.e. the fields are infinite.

On the other hand by using any non-central diagonal element as a parameter, one can define the field $K$ in the group $SL_2(K)$ as follows. Let $t_0$ be one such element. Let $T=C_{SL_2(K)}(t_0) \simeq K^*$ (torus). We may regard $T$ as the group diagonal matrices of determinant 1. There are exactly two subgroups of $SL_2(K)$ of the form $\langle u^T\cup\{1\} \rangle$ for any $1\neq u$ in the subgroup, the strictly upper and lower triangular matrices, say $U$ and $V$ (unipotent) respectively. (Because $x = (1+x/2)^2 - 1^2 - (x/2)^2$ for any $x\in K$, see below). They are both isomorphic to the addive group of $K$. Choose one of them, say $U$. Denote the elements of $T$ by $t(x)$ where $x\in K^*$ and elements of $U$ by $u(y)$ where $y\in K$. Then $T$ acts on $U$ as follows $u(y)^{t(x)} = u(x^2y)$. Thus we get the subfield of $K$ generated by the squares. But since $x = (1+x/2)^2 - 1^2 - (x/2)^2$ for any $x\in K$, the subfield generated by the squares is $K$ itself. Thus the field $K$ is definable with one parameter, namely $t_0$. (Except that the group does not know the unit element 1 of the field, we only get an affine version of a field, something like $K$ with addition and a ternary multiplication $xy^{-1}z$; to fix 1 of the field $K$ we need one more parameter, but this is irrelevant to us). It follows that in the group $SL_2(L)$ both fields $K$ and $L$ are definable.

In particular if the automorphism takes a non-central diagonalizable element of $SL_2(K)$ to a non-central diagonalizable element of $SL_2(L)$, then we will necessarily have $K\simeq L$. This will be so if we can distinguish diagonalizable elements of $SL_2(K)$ from its non-diagonalizable semisimple elements (i.e. diagonalizable in the algebraic closure) in a group theoretic way.