The four-body Coulomb-Born distorted-wave approximation with correct boundary conditions (CBDW-4B) is applied to the K-shell positronium formation from multi-electron atoms at intermediate and high impact energies. In the present approach, both K-shell electrons are treated as active electrons. For collisions of positrons with helium, carbon, and neon atoms, both the post and prior forms of the transition amplitude are calculated and the corresponding differential and integral cross sections are compared with the results of the three-body version of the formalism (CBDW-3B). In order to investigate the effects of the static electronic correlations on the process, initial bound states of the active electrons in helium atoms are described by Hylleraas and Silverman wave functions. Also for positronium formation from helium atoms the obtained cross sections are compared with the available experimental data and also with the results of the other theories.