A question on split-infinitive
I encountered the passage below from a website presented just below.
But to Catholics in the pews, his pontificate is a source of alienation. It is a pontificate, at times, beyond parody: Francis is the first pope to approve of adultery, flirt with proposals to bless gay marriages and cohabitation, tell atheists not to convert, tell Catholics to not breed "like rabbits," praise the Koran, support a secularized Europe, and celebrate Martin Luther.
In the italic part "not to convert" and "to not breed" has the same grammatical role with disparate forms. One with adverb modifier NOT before to infinitive, the other with Not just before infinitive.
Is the difference between two forms just one of styles without any difference of role or is there any profound difference that I don't know about?
(I posed this question because students in Asian areas learning English as a second language are having English grammar tests asking to recognize the error of TO NOT infinitive and to correct it to NOT TO infinitive. But as the sentence in question manifests the sort of test is out of date and inappropriate. Am I right?)