The Veronese surface in $\mathbb{P}^5$ is indeed the only secant defective surface that is not a cone. This is a classical (and non-trivial) result by F. Severi, see p. 6 and Theorem 10.1 in
C. Ciliberto, F. Russo: Varieties with minimal secant degree and linear systems of maximal dimension on surfaces, Adv. Math. 200, No. 1, 1-50 (2006). ZBL1086.14043
ad some of the references cited in the Introduction of the same paper, notably [14, 54, 57].