  In this paper, we present an updated version of our model (KYNXiltr) which
considers thermal reverberation of a standard Novikov-Thorne accretion disc
illuminated by an X-ray point-like source. Previously, the model considered
only two cases of black hole spins, and assumed a colour correction factor
$f_{\rm col} = 2.4$. Now, we extend the model to any spin value and colour
correction. In addition, we consider two scenarios of powering the X-ray
corona, either via accretion, or external to the accretion disc. We use
KYNXiltr to fit the observed time lags obtained from intense monitoring of four
local Seyfert galaxies (NGC 5548, NGC 4395, Mrk 817, and Fairall 9). We
consider various combinations of black hole spin, colour correction, corona
height, and fraction of accretion power transferred to the corona. The model
fits well the overall time-lags spectrum in these sources (for a large
parameter space). For NGC 4593 only, we detect a significant excess of delays
in the U-band. The contribution of the diffuse BLR emission in the time-lags
spectrum of this source is significant. It is possible to reduce the large
best-fitting parameter space by combining the results with additional
information, such as the observed Eddington ratio and average X-ray luminosity.
We also provide an update to the analytic expression provided by Kammoun et
al., for an X-ray source that is not powered by the accretion process, which
can be used for any value of colour correction, and for two values of the black
hole spin (0 and 0.998).
