Definition 2: "to make certain especially by taking necessary measures and precautions"

From the article:

> He sent the final memos to the other board members as disappearing messages, to insure that no one else would ever see them.

> Others were uncomfortable sharing concerns about Altman because they felt there was not a sufficient effort to insure anonymity.

> [...] to insure that the technology was deployed safely

All of these work just fine with that definition of "insure." Your comment that it's "incorrect use of English" is wrong.

The bit you quoted says there’s substantial overlap between the two. The New Yorker style is to prefer “insure” in cases where either could work.