What's the difference between an orbifold with a conical singularity and a conifold?

In Becker, Becker, Schwarz's book 'String Theory and M-Theory: A Modern Introduction', page 360 they explain how an orbifold of $\mathbb{C}/\mathbb{Z}_{2}$ (which is equivalent to $\mathbb{R}_{2}/\mathbb{Z}_{2}$) results in a conical singularity at the fixed point of the orbifold group action.

As I understand it, we could similarly say that $\mathbb{R}_{3}/\mathbb{Z}_{2}$, $\mathbb{R}_{4}/\mathbb{Z}_{2}$,..., $\mathbb{R}_{n}/\mathbb{Z}_{2}$ would all end up with a conical singularity in a similar way, as these examples are simply higher dimensional analogs of the above case.

Also, as I understand it, conifolds are simply manifolds that contain conical singularities.

However, papers such as Klebanov and Witten's 'Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity' appear to distinguish between manifolds orbifolded by $\mathbb{Z}_2$ and conifolds, despite both containing conical singularities.

Is the reason for this that the orbifolds are not simply manifolds with (conical singularities) - they have the additional invariance under $\mathbb{Z}_2$ condition that makes them something more? On the other hand, conifolds are just manifolds shaped so that conifold singularities are present, their lack of invariance under $\mathbb{Z}_2$ distinguishes them from the above cases of orbifold that posses conical singularities.

In other words [orbifold with conical singularities] $\not=$ [manifold with conical singularity (conifold)]

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In most of the string theory literature, 'conifold' refers to a specific type of singularity - an ordinary double point in a threefold. This is locally (analytically) given by the equation $$x y - w z = 0~.$$