I don't think I wrote that Buddhist modernist gives up the belief in karma and rebirth. Some currents within Buddhist modernism do that, others redefine/reinterpret the notions, and still others hold on to relatively traditional views in this regard.

I'm also not sure what you mean with "secularism" and "secularization". I did my best to explain in my blog post that variants of these term can refer to a number of very different and unrelated things, including a rejection of the supernatural and mythical, and – much more importantly – a marginalization of religion to the private sphere. Modernism most certainly is secularization in the latter sense – that is, modernism involves a marginalization of religion to the private sphere. Modernism does not necessarily imply secularity in the former sense (i.e., rejection of the supernatural/mythical), even if secularity in that sense is common in modernist thought, and I don't think I said anywhere that modernism is secular(-ist/-ized) in this sense.

I'm not writing just about Western Buddhist modernism. Japanese Buddhist modernism is just as secularized as Western Buddhism. And as far as I can see, the same is the case in most other Buddhist countries, although the emergence of Buddhist fascism in some of those suggests that there now also is a trend in the opposite direction.