When I select Search box I want it to default to Apple's search rather than my collection. For me that has been a pain point. Clearly from your case, various users respond differently to features.
It will default to whichever you used last, and it's a single tap to switch. I'm not sure how it could be better designed. Two different search buttons seems like a bad idea. Ignoring your last-used to always focus on Apple's preferred option seems like a bad idea.
Last-used with a single tap to switch seems ideal to me.
Last-used with a single tap to switch seems ideal to me.