So if a Jew converts to Christianity is not a Jew? What is he?
A Christian. If the convert drops the religious traditions of Judaism, the food laws, sabbath, etc, the convert no longer has an identification of being a Jew. One could be considered a cultural Jew but this is again a construct since the convert no longer practices strictly Jewish traditions if any. The cultural identification is dictated by a religious view or former in this case, which just is more evidence for it being a religious view not ethnic grouping. Besides in the OT it is crystal clear Jew is the identification of the post-Babylonian exiles which claim to maintain the true religion. This was done in order to show that they, not the Samaritans, are the true believers. The term by it's definition already excludes a similar belief system of the Samaritans. The Khazars are converts which were ethnic Turks of Central Asia. They have no ties with the Hebrews or Israelites. They only have a connect with Judaism post-Babylon by conversion, hence it's a religious identification.
There are atheists Jews, agnostic Jews, Christian Jews. Khazars could have been now another ethnic group related to Judaism, but it didn't worked out. Of course it's an unique situation that Jewish people are now the only ethnic group related to Judaism, so you can have it both ways, Jews as followers of Judaism and Jews as the descendents of Israeli tribes from Judea. Yes, it's easy to conflate this 2 things.
A Jew is defined by a religious view as per above. Atheist Jews are just former Jews that changed religious views. Again the ties are to former religious views not an ethnic grouping. It conflates nothing since Jew as an ethnic identification is a PC construct. No more than ethnic Christian. See the double-standard at play here? Do you accept the identification of ethnic Muslim?
As we speaking about laws, as far as the law regarding Holocaust denial is made in my country (and all Europe I think), it is dealing with racism, there is nothing about Judaism. There have been many ethnic groups killed in Holocaust, not only Jews.
The Holocaust was based on racial purity. However since race is a construct it's views have no merit in reality. It has merit in history as an example of people embracing false ideas.
This all thing about specific laws made for Jews only feed Muslim apologists who like to play the victim card.
Likewise people can merely point out the the specific laws between Muslim and non-Muslim in current and past nations. Such apologist arguments merely point out favoritism, rightly so, and respond with hypocrisy.