In a previous paper [M. Kundu, Phys. Plasmas 21, 013302 (2014)], fractional collisional absorption (α) of laser light in underdense plasma was studied by using a classical scattering model of electron-ion collision frequency ν(ei), where total velocity v=√[v(th)(2)+v(0)(2)] (with v(th) and v(0) as the thermal and the ponderomotive velocity of an electron) dependent Coulomb logarithm lnΛ(v) was shown to be responsible for the anomalous (unconventional) increase of ν(ei) and α(∝ν(ei)) with the laser intensity I(0) up to a maximum value about an intensity I(c) in the low temperature (T(e)<15eV) regime and a conventional ≈I(0)(-3/2) decrease when I(0)≫I(c). One may object that the anomalous increase in ν(ei) and α were partly due to the artifact introduced in lnΛ through the maximum cutoff distance b(max)∝v. In this work, we show similar anomalous increase in ν(ei) and α versus I(0) (in the low temperature and underdense density regime) with more accurate quantum and classical kinetic models of ν(ei) without using lnΛ, but with a proper choice of the total velocity dependent inverse cutoff length k(max)∝v(2) (classical) or k(max)∝v (quantum). For a given I(0)<5×10(14)Wcm(-2), ν(ei) versus T(e) also exhibits so far unnoticed identical anomalous increase as ν(ei) versus I(0), even if the conventional k(max)∝v(th)(2) or k(max)∝v(th) (without v(0)) is chosen. The total velocity dependent k(max) in the kinetic models, as proposed here, is found to explain the anomalous increase of α with I(0) measured in some earlier laser-plasma experiments.