In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graphs with maximum degree $Δ= 4$ and girth at least $4$ are $2$-distance list $(Δ+ 7)$-colorable and injectively list $(Δ+ 5)$-colorable. Additionally, we prove that planar graphs with $Δ= 4$ are injectively list $(Δ+ 7)$-colorable and exact square list $(Δ+ 6)$-colorable.