Counterexample: loop of 8 nodes, numbered 0, 1, ..., 7.
Start with one pointer at node 0, moving with speed 5. Start the second pointer at node 1, with speed 3 which is coprime to 5, the speed of the first pointer.
The pointer positions at each step are:
0 5 2 7 4 1 6 3 repeat
1 4 7 2 5 0 3 6 repeat
The condition we need in order to guarantee it works regardless of where the two pointers start is that the difference between the speeds is coprime to the cycle length.
Start with one pointer at node 0, moving with speed 5. Start the second pointer at node 1, with speed 3 which is coprime to 5, the speed of the first pointer.
The pointer positions at each step are:
The condition we need in order to guarantee it works regardless of where the two pointers start is that the difference between the speeds is coprime to the cycle length.