This semester teaching undergraduate linear algebra, I defined a pivot of a matrix as:

a one with nothing but zeros to the left, nothing but zeros above, and nothing but zeros below.

Some textbooks don't require a pivot to be a one, just to be nonzero. But in any case, the books I've seen state the requirement about surrounding zeros in more complicated way.

The same day, I gave definition of reduced row echelon form simpler than what I'd seen in textbooks.

A matrix is in RREF if every nonzero row contains a pivot, the pivot rows are above the zero rows, and the pivots descend to the right.

Maybe I just haven't been reading the right textbooks. I was given a review copy of Gareth Williams' textbook, but too late to use it this semester. It doesn't define pivots, but its definition of RREF matches mine, with instances of "pivot" expanded into a definition of pivot that matches mine.