New arguments are given here in favor of Weibel-type instabilities as one of the most plausible sources of the cosmological magnetic field. The Weibel instability has recently been proposed as one of the secondary mechanisms of relaxation for the large interpenetrating formations of galactic and intergalactic plasma. Here, these investigations are extended to counterstreaming plasmas which have, in addition, intrinsic temperature anisotropies, and where any form of the Weibel-type instability can be excited. This can be a simple filamentation instability due to the relative motion of counterstreaming plasmas, or a Weibel-like instability when it is generated by an excess of transverse temperature with respect to the streaming direction. But it can also be a cumulative filamentation/Weibel instability when the plasma is hotter along the streaming direction. Such plasma systems are relevant for the relative motions of filaments and sheets of galaxies, and are expected to exist at large scales and any age of our Universe. For such counterstreaming plasmas with internal temperature anisotropies, any Weibel-type instability mentioned before can become the primary wave relaxation mechanism of the plasma anisotropy, because it develops easily faster than the principal competitor, which is the two-stream electrostatic instability. The estimations made here for typical parameters of intergalactic plasmas, provide micro-Gauss levels of the magnetic field of Weibel type, which are consistent with magnetic field values, 10–7-10–5 G, derived from Faraday rotation measure of the linearly polarized emission of galactic or extragalactic sources.