  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 $\Delta = 4$ and girth at least $4$ are $2$-distance list
$(\Delta + 7)$-colorable and injectively list $(\Delta + 5)$-colorable.
Additionally, we prove that planar graphs with $\Delta = 4$ are injectively
list $(\Delta + 7)$-colorable and exact square list $(\Delta + 6)$-colorable.
