How is the order determined? I thought it would be the same as the draft order of the teams signing the FA, but that obviously is not the case.

I thought this was if we wanted a Reuben or ham and swiss

So apparently no one knows? I think MLB assigns the picks based on the value of the player lost, though I'm not sure.

Given that we have a actual ability ratings for ranking players as opposed to RL performance ratings which are shaped by park and style of play factors, that seems the most obvious way for HBD to handle it. However, that doesn't seem to be the case.

Any ideas?

It is same as draft order, it just doesn't reset until the draft prospects become available, as that is the deadline for compensation. Once the prospects are available to see, the order locks.