OP answering own question.
According to the most recent (2022-03) statement I could find, the 2022 US Nuclear Posture Review (NPR) has been submitted to Congress but not yet declassified. The DoD media release says,
the President [Biden] has articulated his vision for U.S. nuclear deterrence strategy: As long as nuclear weapons exist, the fundamental role of U.S. nuclear weapons is to deter nuclear attack on the United States, our allies, and partners. The United States would only consider the use of nuclear weapons in extreme circumstances to defend the vital interests of the United States or its allies and partners.
In a 2022-09-25 interview (transcript here) with CBS's "Face the Nation", US National Security Advisor Jake Sullivan said the following.
We have communicated directly, privately, at very high levels to the Kremlin that any use of nuclear weapons will be met with catastrophic consequences for Russia
If Putin took 'catastrophic consequences for Russia' as clearly implying the use of strategic nuclear weapons against the Russian heartland in the event that Russia used tactical nuclear weapons in Ukraine, which is 'only' a US partner or ally, perhaps this would give the provocative sense Putin seems to have in mind.
ETA: @Fizz rightly points out that Sullivan's interview occurred after Putin's speech, so Putin couldn't have been referring to the interview, but perhaps could have been referring to the private communications Sullivan mentions, if those occurred before Putin's speech, which we don't know.
On the other hand, if Putin had read such a meaning into US DoD statements on US nuclear posture, Sullivan's statement indicates that such a reading was justified.
In a 2021-12-07 interview with Fox News, US Senator and Senate Committee on Armed Services member Roger Wicker said, contemplating the prospect of a Russian invasion of Ukraine,
We don't rule out first-use nuclear [by the US against Russia].