If two points $P,Q$ which are not umbilical are given, then there are in general several geodesics but only one that is shortest among them all. Their $ u^{'}, v^{'}$ are different, but unique for any one of the geodesic choices.
If $P,u^{'}, v^{'}$ are given there is no guarantee that the line so defined passes through $Q.$
These can be physically verified between two points on a cylinder/cone using a taut thread for example.