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Arguable, the two definitions of correlation are just system-specific versions of the same meaning: the degree of dependence that bodies in a system have with each other. In classical mechanics, this translates as the difference between the sum of independent solutions and the exact solution. In quantum mechanics, the fermionic nature of the bodies in the system causes there to be an independent particle solution that isn’t just a summation/product (the Hartree-Fock solution) and so two levels of difference arise: the difference between two possible independent-particle solutions (exchange), and the difference between the quantum-adjusted independent-particle solution and the exact solution (correlation).
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