I'm studying differential geometry to have a good writing in a paper and contextualizing the reader about this topic in Applied Mathematics context. In this context, I'm applying the Theorem 2.1 to some optimization problems with analytic constraints. Basically, the context of manifold I'm using is in the finite-dimensional, Hausdorff, second countable manifold of pure dimension (i.e. such that all the connected components have the same dimension). All my manifolds considered is of class $C^{\omega}$, i.e. real analytic. Here, my concern is with clarity to the reader, which is not really fluent with this tool. For example, seems hard to me to understand and grasp the differences between several differentiable manifolds definitions, and which one I should state for the reader.
As an example, the Varadarajan's book says that a manifold is a pair $(M,\mathfrak{D})$ satisfying the following:
.
While on book Narasimhan's Book I found a simpler one without introducing a differentiable structure:
But there is another third and more geometric one I have found on Munkres's book:
My question is:
Is all these definitions equivalent when translated correctly? What are the differences? Which one I should state for the reader first?