There are way too many approaches to ODEs to have any one book cover them all. I occasionally use a book called Differential Equations and Dynamical Systems, by Lawrence Perko. The focus of this book is on qualitative behavior - existence of fixed points, limit cycles, blow-up solutions, etc.

I would not call this a standard introduction to ODE - it does not cover some of the absolute basics. However, I think the emphasis of this text on geometry, and on using some more modern results, makes the book a decent choice.

Some flaws: The book really only presupposes mastery of analysis. There are some tools missing, in particular from geometry/topology, that could make the presentation a bit cleaner. It sounds like you have a strong geometry/topology background, so maybe this disqualifies this text for you.

For a more classical treatment of ODEs, in particular the treatment of ODEs as linear operators (Sturm-Liouville theory), I might go for Coddington's Theory of Ordinary Differential Equations. It is very classical, but it really does cover all the essential theory.