New York Times v. Sullivan was a defamation case. The 1st Amendment protects your right to show reckless disregard for truth or falsity, unless you're slandering someone.
Do you have a citation for your second sentence? NYT v. Sullivan was a defamation case, but my assumption is that the level of protection the First Amendment provides to any given act of speech isn't affected much by which law is trying to penalize that speech. I could easily be wrong; is there a case where the issue came up? (Also, Elon's tweet may count as commercial speech, which is less protected in general.)
I'm basing the second sentence in the narrowness of the exceptions to freedom of speech in the US.
If the government were to try to impose restrictions on "untrue" speech in areas outside defamation, particularly in politics, I would view that as a major assault on the 1st Amendment.
