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.