Tricky one, Tommy!
The participial phrase "Seeing the rain come on" is, grammatically, an adjective phrase modifying the main subject "we." Strictly speaking, therefore, the sentence is simple.
But the plot thickens, my friend. A participial phrase is a non-finite subordinate clause equivalent = Because we saw the rain coming on (= a causal clause). It thus falls between two stools. Your textbook is half-right and half-wrong due to the hybrid nature of a participle.
Good luck!
Peter C.
Milton MA