Definition of twisted geometries and existence of coordinate transformation for twisted $AdS_2 \times S^2$ In the paper Multiply Twisted Products by Yong Wang, general definitions for so called warped and twisted products are given:
A (singly) warped product $B \times_b F$ of two pseudo-Riemannian manifolds $\left(B\,,g_B\right)$ and $\left(F\,,g_F\right)$ with a smooth function $b\,:\,B \to \left(0\,, \infty \right)$ is the product manifold $B \times F$ with the metric tensor $g = g_B \oplus b^2 g_F$. We call $\left(B\,,g_B\right)$ the base manifold, $\left(F\,,g_F\right)$ the fiber manifold and $b$ the warping function.
A twisted product $\left(M\,,g\right)$ is a product manifold $M = B \times_b F$, with a smooth function $b\,:\,B \times F \to \left(0\,, \infty \right)$ and the metric tensor $g = g_B \oplus b^2 g_F$.
The notion of warped products seems to be a rather standard definition, first introduced by O'Neill. I wonder if the definition of twisted products (and the generalizations of warped an twisted products) in the paper given above is also standard. Many theoretical physics papers sometimes talk about warped and twisted spacetimes in a rather handwavy way, so I am never really sure if they are talking about a mathematical definition or they just want to sound fancy. 
For example, Compère talks in an overview paper about the Kerr/CFT correspondence about the following metric 
\begin{align}
  \mathrm{d}s^2 = J\left(1+\cos^2\theta\right)\left[-r^2dt^2 + \frac{dr^2}{r^2} + d\theta^2 \right] + \frac{4 \sin^2\theta}{1+\cos^2\theta} \left(d\phi + rdt\right)^2\,,
\end{align}
being a " warped and twisted product of $AdS_2 \times S^2$ " (J is just scaling the metric). This seems to be a different definition of a twisted product, as in Wangs paper twisted is a generalization of warped. Furthermore I can't neither make out the above metric to be $AdS_2 \times S^2$, nor that it is for $\theta = \frac{\pi}{2}$ a "twisted product of $AdS_2$ and a circle of constant radius" (as this paper suggests), as the suggested geometry should not have off-diagonal terms like $\mathrm{d}\phi \mathrm{d}t$ in the metric. For clarity, the metric of $AdS_2$ in Poincare coordinates is
\begin{align}
    \label{eq:poincarepatch}
    ds^2 =  - r^2dt^2 + \frac{dr^2}{r^2}\,,
\end{align}
where we set the curvature constant to $1$ to match the above expression. 
My questions are:


*

*Am I missing coordinate transformations that make this obvious? As mentioned below this would have to make the metric diagonal.

*What is a standard definition for a twisted product of Pseudo-Riemannian manifolds?


EDIT: So as some comments say, the definition of a twisted product seems to be canonical. I need a coordinate transformation that gets rid of the off-diagonal term or at least a reference to a theorem that ensures the existence of such a coordinate transformation.
 A: The short answer is "what physicists mean by 'warped' and 'twisted' geometry" is not the same as "what differential geometers mean by 'warped' and 'twisted' geometry". The use is a lot more qualitative and a little loosey-goosey, but on the other hand very easy to visualize. 
From the physicists' point of view, a twisted geometry is basically something that is not diagonal. The idea is basically that you can consider "nice" manifolds that admit local fibrations with the fiber $F$ orthogonal to the base $B$, so locally the geometry looks like $g_B \oplus g_F$, where here we do allow both $g_B$ and $g_F$ to depend on both $B$ and $F$ coordinates. 
A  "twisted" version of this geometry would be one that still has the fibration, but you now "twist the fibers around the base", so now that the fibers are no longer orthogonal to the base. 
A classic example in relativity is that of the Kerr(-Newman) space-times. The idea being that an "untwisted" black hole is something like the Schwarzschild solution, which has nice factorization as some base manifold that is orthogonal to the integral curves of the time-like Killing vector field. The rotating black holes on the other hand have the time-like Killing vector field "twisted" around the base, losing the orthogonality. 
The nomenclature of "twist" came from the examination of geodesic congruences. The "vorticity" tensor is also called the "twist" tensor, and its vanishing is (essentially by Frobenius' theorem) equivalent to the local integrability of the orthogonal hyperplane distribution to a vector field. 
Next, from the physicists' point of view, a warped geometry is something that is deformed from a reference geometry by squashing/stretching factors. (So the notion of warping is closed to the notion of a "warped product", but not the same.) Basically you start again with a nice manifold that admit local fibrations as $B\times F$, with the geometry $g_B \oplus g_F$. A "warping deformation" of this geometry is one that replaces the geometry by $ \lambda g_B \oplus \lambda^{-1} g_F$ so that the new geometry looks like the old one but stretched in some directions but squashed in some other directions. 

As an example: from the paper of Compere that you linked to (pg. 15), you see that he (like many physicists) think of $\mathrm{AdS}_3$ as the fibered manifold with base $\mathrm{AdS}_2$ and fiber $\mathbb{S}^1$, but with the fiber "twisted around" the base. 
Insofar as "warping": you can see this survey which uses the common physics terminology to discuss what is meant by "warped $\mathrm{AdS}_3$". 
