Understanding how to find a basis for the row space/column space of some matrix A.

yes you're correct.

note that row echelon form doesn't necessarily result in 'leading 1s'. it's 'reduced/canonical row echelon form' that requires that form.

having reduced your matrix to the set of the linearly independent rows/columns via the row transformations, you can choose either the new reduced vectors with leading pivots (1s or otherwise), or the corresponding vectors from the original matrix*. they are effectively 'the same'. i'd go with the reduced vectors however, as they make any further manipulation or plotting easier

*see caveat raised by user84413