I am relaxed. I thought your question made sense so I wanted to make sure I understood it. But I don't understand exactly how the law works in this situation.
Registration was never a law in the past. It was a rule. There was much debate about whether it was legally required or should be. Many refused to register. AMA put it off for some time. It was eventually challenged in court. It was vacated. Now, it seems the court's decision has been undone and the registration "rule" has been restored. But if it was restored, it is what it was before.
So does the President's signature make it law? After all, as you've pointed out, the language in Sec 336 is unchanged.
Or is it still a rule?
Does it matter?
I suppose it should be regarded as a law because if it was ever challenged again, the challenge would fail due to historic precedence.