Question

What's the physical meaning of the eigenvalues of the spin-flipped density matrix?

Answer 1

The expansion in (1)(1) isn't really a definition - it is just one possible mathematical expansion that, depending on context, may or may not have physical significance. The eigenvalues & eigenvectors do admit the interpretation of a probabilistic mixture of the given states with the given probabilities provided by a purely classical RNG. However, this is not the only way to produce density matrices, and you can get mixed states by e.g. tracing out one half of an entangled system that's otherwise in a pure state.