Resetting Android system / Sound picker default

I was playing around with different ringtones.

Until now, when I went to choose a new ringtone, I was offered the choice of using "Sound picker" or "Android system" to do so.

I have just set "Android system" as the default by ticking the box in the popup, but now I realize it doesn't give the same choices as "Sound picker".

But I can't clear the default setting, when choosing a new ringtone it goes straight to the "Android system" choices without giving me the choice to use "Sound picker".

Usually you can reset this by going into settings->Apps and clearing an App's defaults. But "Android system" isn't listed as an App.

There must be a setting somewhere else. Any idea where?