The explanation you found in Duden is correct, but it is very abstract, and so I am not surprised that you have problems understanding it. Indeed, seeing more examples could be helpful:
Auf Initiative seiner Mutter bewarb sich Heinz beim Bäcker als Geselle.
On (!) his mother's initiative, Heinz applied for an apprenticeship at the baker's.
Auf Veranlassung des Betriebsrats befasste sich die Werksleitung mit dem Vorfall.
At the instigation of the works committee, the works management concerned itself with the incident.
Auf einen Brief antworten.
To answer "on" a letter / to answer a letter.
More examples :
Aufgrund des schlechten Wetters...
Due to the bad weather... ("On grounds" of the bad weather...)
Auf deinen Rat hin habe ich mir den grünen Pulli gekauft.
On your advice I bought the green pullover.
(I don't know if you would say that so, with "on" in English, really, but I think it still makes the meaning, clear, doesn't it?)
Auf Ersuchen des Dorfvorstehers gestattete der Landrat den Festumzug.
On request of the village headman the governour gave permission for the pageant.
Learning German: better memorize expressions
If what you are doing is learning German, I would doubt that you have much use of learning theoretical categorizations of meanings (here for the preposition "auf"). I think it will be more useful to memorize some concrete and frequently used expressions in some context, and use them actively.
Note that expressions like auf Veranlassung seiner Mutter, auf Initiative des Pfarrers and so on are from a rather bureaucratic register of speech. People do not speak like this in everyday communication. Rather you can find such expressions in formal letters, like business letters.
Expressions for everyday situations:
auf Rechnung = to buy not cash but on (!) account
auf Rezept (Medikamente kaufen)
auf deinen Rat hin
auf eigene Gefahr
auf eigene Rechnung
auf Gedeih und Verderb
auf Teufel komm raus
auf gut Glück
auf Schritt und Tritt
auf Heller und Pfennig (= einen Geldbetrag sehr genau ausrechnen oder auszahlen)
In all these cases (or idioms) you can argue - as Duden would do - that here the auf introduces some condition under which the action takes place. But knowing this does not help you (nor a native speaker). What helps is knowing the expressions as such, and their meaning and typical use-cases.
If anybody feels competent to improve my English translations: welcome!