Musk has been promoting extreme right-wing opinions, conspiracies and outright lies for quite a while now and he's obviously seeing the Trump presidency as a key success in his plot for world domination, so he no longer needs to hide his fascism/racism etc.
Why other organisations are seeking to downplay actual Nazi salutes must be because they are scared of the power imbalance and seek to appease (hint: that never works).
I don't see how Musk's two Nazi salutes can be defended when he deliberately did it twice and has a history of supporting extreme right-wing views.
Once you start differentiating between Zionist interests and Jewish interests, the last 80 years (and especially the last 15 months) makes a lot more sense.
They know which way Israel's bread is buttered. Trump just removed all sanctions for the settlers, and allowed the sale of significantly bigger bombs to Israel on his first day.
Israel has a huge amount to gain from a Trump administration. ADL isn't going to blow that opportunity by calling out Elon's Nazi salute.