Here is a very short elementary proof, but perhaps not completely trivial.

Assume there $n$ rows and $m$ columns. Represent the traversing path by a line, going from the center of one unit square to the center of the next and so on.

There are then three (overlapping) types of unit squares with path-segments, according to the Venn diagram pictured (notice that end-of-path squares are considered CORNERS):

[![Unit square types][1]][1]

**Trivial observation**: any row of the rectangle containing a HORIZONTAL unit square must contain at least $2$ CORNER squares. Likewise any column containing a VERTICAL unit square must contain at least $2$ CORNER squares.

**Conclusion**: if each row contains at least one HORIZONTAL square, then there are at least $2n$ CORNER squares. Otherwise there is one row consisting entirely of VERTICAL squares, in which case each column contains at least one VERTICAL square, and then there are at least $2m$ CORNER squares.

Since $2$ of the $2n$ or $2m$ CORNER squares are end-of-path squares, there are at least $\min(2n-2,2m-2)$ true corners, as expected.

  [1]: https://i.sstatic.net/WxtJgVuw.png