Early B-type stars are invaluable indicators for elemental abundances of their birth environments. In contrast to the surrounding neutral interstellar matter (ISM) and HII regions their chemical composition is unaffected by depletion onto dust grains and by the derivation of different abundances from recombination and collisional lines. In combination with ISM or nebular gas-phase abundances they facilitate the dust-phase composition to be constrained. Precise abundances of C, N, Mg, Ne, Fe in early B-type stars in the Orion star-forming region are determined in order to: a) review previous determinations using a self-consistent quantitative spectral analysis based on modern stellar atmospheres and recently updated model atoms, b) complement results found in Paper I for oxygen and silicon, c) establish an accurate and reliable set of stellar metal abundances to constrain the dust-phase composition of the Orion HII region in Paper II of the series. A detailed, self-consistent spectroscopic study of a sample of 13 narrow-lined B0V-B2V stars in Ori OB1 is performed. High-quality spectra obtained with FIES@NOT are analysed using a non-LTE method and line-profile fitting techniques, validating the approach by comparison with results obtained in Paper I using line-blanketed non-LTE model atmospheres and a curve-of-growth analysis. The two independent analysis strategies give consistent results for basic stellar parameters and abundances of oxygen and silicon. The extended analysis to C, N, Mg, Ne, and Fe finds a high degree of chemical homogeneity, with the 1sigma-scatter adopting values of 0.03--0.07 dex around the mean for the various elements. Present-day abundances from B-type stars in Ori OB1 are compatible at similar precision with cosmic abundance standard values as recently established from early-type stars in the solar neighbourhood and also with the Sun. (abridged)
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