Well, I have absolutely no idea why I thought the original question was about reordering fields in form views rather than expanded records. (I see my reply was written at 6:20 a.m., so maybe I was waiting for the coffee-maker to finish.)
The order of fields (and which fields are hidden) in an expanded record is determined by their order in the current view.
The order of fields (and which fields are hidden) in a linked record reached by ‘drilling through’ from another record is determined by their order in that table’s default view — that is, the table’s first or upper-most listed view. However you configure that view — regardless of whichever type of view it may be — determines the visibility and order of fields when accessed as a linked record.
The drag-and-drop method I so confusingly mentioned applies only to ordering fields in a form view: