The New York state governor announced the statewide lockdown on March 20 (to begin two days later, as indicated above). The statewide lockdown is not causally responsible for social distancing before then.
The same is true for the announcement of closing New York City public schools, which came on March 15. If private schools closed before then, the closure of public schools later cannot "take credit" for it.
This distinction is important for causal modeling, both of the past as well as the future, if/when orders from authorities are lifted. The effectiveness of the lockdown order is measured against a world without the lockdown order, which is still a world with a lot of social distancing.
Telling people "we recommend physically distancing and wearing masks" does not qualify as lockdown. I hope that I made that clear in my writeup.
Thanks for jumping in here, I think you made the point far better than I could have.