This is a question about cohomogeneity one actions of a compact Lie group $G$ on a Riemannian manifold $M$, such that $G$ acts via isometries. Many articles have been published about classifications of such actions. Different articles classify actions up to different notions of equivalence. Here are two examples:
Classification up to $G$-diffeomorphism (i.e. there exists a diffeomorphism $M \rightarrow M$ that is $G$ equivariant, i.e. $f(g \cdot_1 x)=g \cdot_2 f(x)$, where $\cdot_1$ and $\cdot_2$ denote two different actions of $G$ on $M$): Karsten Grove, Burkhard Wilking & Wolfgang Ziller: POSITIVELY CURVED COHOMOGENEITY ONE MANIFOLDS AND 3-SASAKIAN GEOMETRY
Classification up to isometric orbit equivalence (i.e. there exists an isometry $f:M \rightarrow M$ such that $G \cdot_1 f(x)=f(G \cdot_2 x)$): J. Berndt M. Domínguez-Vázquez: COHOMOGENEITY ONE ACTIONS ON SOME NONCOMPACT SYMMETRIC SPACES OF RANK TWO
The notion of "$G$-isometry" (replace the diffeomorphism in point 1 by an isometry) would be the most natural to me, but I haven't found it in the literature.
Question 1: Are these notions identical?
Question 2: One application of a classification of cohomogeneity one actions was to find an exotic nearly Kähler structure on $S^6$ (Lorenzo Foscolo, Mark Haskins: New $G_2$-holonomy cones and exotic nearly Kähler structures on $S^6$). They used a classification of cohomogeneity one actions on $S^6$ up to $G$-diffeomorphism. Could it happen that there is another, hidden exotic nearly Kähler structure on $S^6$ which arises from a $G$-diffeomorphic cohomogeneity one action on $S^6$ that is not $G$-isometric? (That would lead to two nearly Kähler structures that are not isometric)