 Solution

For $K<0$ there is at most one. Two essential simple closed geodesics cannot form a geodesic bigon by Gauss-Bonnet theorem, so they are disjoint; the annulus between them would have $\chi =0$ and geodesic boundary, forcing $\int K=0$, a contradiction. Thus multiple geodesics in the flat case are disjoint, while in the positive example they need not be.