Apple's round checkboxes - Where and when are they used?

In general circular selection boxes should also be fine, given that there is no use case where there is a combination of multi select items along with single selection items, example in a schedular where a person can select multiple days but can only have single selection of exact time. In these cases having a traditional square shape for checkbox and circular one for radio button will be easy for the user to comprehend.

You mention the Apple references but it seems like you might not even be designing for iOS, or at least are looking more for UX advice in general than for the preferred choice on iOS apps.

Using my pretty new Android phone, if I open my text messages and hold-press on an item it selects it and shows a checked circle on the left just like in your Apple images.

I've deleted messages using this before, and as a user perspective I didn't find it confusing because I had first long-pressed on a row, and then the circles appeared with the 1 from the row I selected highlighted. I triggered it so it was obvious what it meant.

If I navigated to a new page and there were circles all down the side from the start I might think it was a radio list, but because they appear as a result of my action it seems obvious that I can choose multiple.

Anyway I think this is not only used by Apple, so if you are looking for some reference to show that Apple only uses them in certain circumstances, well Android uses them too so it seems more universal than that.

My understanding of Apple's design philosophy is to design products that can be easily used by idiots (because if idiots can use them, smart people can too, so it means everyone can use them). Contrast this with Microsoft's design philosophy of designing for middle-of-the-range people which can be difficult for idiots to use, and the Linux design philosophy where you require a PhD to connect to the WiFi.

Smart people can use layouts designed for idiots too, so it makes sense to do it this way.

Anyway for this I think it won't be confused as a radio list if the user's action triggers the circles to appear.

I think one could only argue that round checkboxes would be confusing to users that actually understand the differences between checkboxes and radio buttons, most users don't. I don't believe the average user relies on shapes to know whether or not they can select multiple items. And I don't think users would actually be confused, rather they'd just feel a subtle annoyance at the inconsistency.

While I personally prefer my checkboxes square this might be more of a design decision than anything else.

If you want to make a strong argue towards square you might need to find design guidelines or conventions already in place in your app and argument for consistency. Consistency is an important and can often make a strong case towards one decision over another.

From my experience using Apple devices, I believe using this round type of checkbox has been around for a while when selecting multiple items on iOS. I was able to find this documentation regarding multiple selection. If the app is going to be on iOS devices, I would say round checkboxes are the way to go for multiple selection.