[en] This paper together with the previous one (Borowiec, Eur. Phys. J. C 48:633, 2006) presents a detailed description of all quantum deformations of the D=4 Lorentz algebra as a Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra o(3,1), obtained by twisting of the standard q-deformation U_q(o(3,1)). For the first twisted q-deformation an Abelian twist depending on the Cartan generators of o(3,1) is used. The second example of twisting provides a quantum deformation of Cremmer-Gervais type for the Lorentz algebra. For completeness we describe also twisting of the Lorentz algebra by a standard Jordanian twist. By twist quantization techniques we obtain for these deformations new explicit formulae for the deformed coproducts and antipodes of the o(3,1)-generators. (orig.)