Ambiguity I - Collinear Points
So far, GGB's Conic[] tool and I agree.
If the coefficient matrix has two free parameters, it would appear safe to conclude that the conic section is not uniquely defined.
But this is not so.
There are two cases that cause ambiguity:
I. four or five points are collinear.
II. two or more points coincide.
Let me tackle case I, as it can be resolved with relative ease. Consider five points in space: