People can be weird about this sort of thing, especially with Apple. I remember when the iPhone XR came out some people would insist on always writing "iPhone Xʀ", for example.

I believe these forms were used on startup (i.e. they were in the roms in text form this way).

As I recall, all print publications of the era used ][ and //

I remember typing it this way myself on the early BBS scene. Personally my system was a Radio Shack TSR-80 Model III. (That is III not 3)

I have an Apple II plus original reference manual here, and it uses “Apple ][“ (brackets) only on the cover. In the book text, it uses “Apple II” (Roman numeral) throughout.

No, they used it elsewhere. When the machines booted, they displayed "Apple ][, "Apple //c", etc. at the top.