  The detailed investigation of the Higgs sector at present and future
colliders necessitates from the theory side as precise predictions as possible,
including higher order corrections. An important ingredient for the computation
of higher order corrections is the renormalization of the model parameters and
fields. In this paper we complete the renormalization of the 2-Higgs-Doublet
Model (2HDM) Higgs sector launched in a previous contribution with the
investigation of the renormalization of the mixing angles $\alpha$ and $\beta$.
Here, we treat the renormalization of the mass parameter $m_{12}^2$ that softly
breaks the $\mathbb{Z}_2$ symmetry of the 2HDM Higgs sector. We investigate the
impact of two different renormalization schemes on the sample Higgs-to-Higgs
decay $H\to hh$. This decay also allows us to analyze the renormalization of
the mixing angles and to confirm the properties extracted before in other Higgs
decays. In conclusion we find that a gauge-independent, process-independent and
numerically stable renormalization of the 2HDM Higgs sector is given by the
application of the tadpole-pinched scheme for the mixing angles $\alpha$ and
$\beta$ and by the use of the $\overline{MS}$ scheme for $m_{12}^2$.
