The determinant of a 2x2 matrix can be calculated by multiplying the components of the main diagonal, and subtracting the product of the off-diagonal.
The trace is found by summing the components of the main diagonal.
I have encountered a formula that involves summing the main diagonal, and subtracting the sum of the off diagonal. This seems to closely represents the formula for the determinant of a 2x2 matrix as well as the formula for the trace.
Is this "called" something, or connected to these two concepts in any way?
One of the most important properties of the determinant and the trace is invariance under conjugation, meaning that for any invertible matrix $Q$, and any matrix $A$, we have $$ \mathrm{trace}(QAQ^{-1}) = \mathrm{tr}(A)$$ and $$ \det(Q A Q^{-1}) = \det(A)$$
If we try out your function $f$, $$ f \begin{pmatrix} a & b \\ c & d \end{pmatrix} = (a + d) - (b + c)$$ we can see that it is not preserved under conjugation, since for example $$ f \left( \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1/2 & 0 \\ 0 & 1 \end{pmatrix} \right) = f \begin{pmatrix} a & b/2 \\ 2c & d \end{pmatrix} = (a + d) - (b/2 + 2c)$$ so I think this function $f$ probably is not related to the trace or the determinant in an obvious way, and certainly not a "natural" one.