Let $f:M \to N$ be a smooth map of a closed $n$-dimensional manifold $M$ into a $p$-dimensional manifold $N (n \geq p)$. When $f$ has only definite fold singular points, we call it a special generic map. Porto and Furuya introduced the notion of regular equivalence for such maps. In this paper, we define another equivalence relation for special generic maps, called weak regular equivalence and we classify special generic maps up to these equivalences in the case where $M$ is a connected orientable closed $n$-dimensional manifold with $n \geq 3$ and $N$ is the plane. We also show the existence of a pair of special generic maps which are $C^0$ right-left equivalent but are not $C^\infty$ right-left equivalent.