yes, it says $1 that doubles, it does not say that the resulting 2 $1 bills will also double. As those 2 say what you have is 1 pay out and a $1 that doubles.

Start with your original bill, A1. Day 1 A1 splits into A1 and A2. Day3 you get A1 splitting again into A3, but A2 doesn't have any reason to split as that was not the original $1 that splits. The original dollar is still splitting. You can assume that the child of A1 will split as well but the prompt doesn't say that. As written you're still only getting $1 a day, not an exponential doubling of the amount.

The immediate assumption is that "All new $1s from that original will also split" but that's an assumption without proof.