In its alphabet (in primers and grammar books), broad and regular shapes of О/о share the same position, as they are not considered different letters.
Uppercase is typically represented by broad Ѻ, and lowercase is either regular о or dual: both broad ѻ and regular о (in the same way as Greek uppercase Σ is accompanied with two lowercases σ, ς).
In standard Church Slavonic orthography (since the middle of the 17th century until present time), the broad shape of letter On is used instead of the regular shape of the same letter in the following cases: Historically, Broad On was also used in the later Old Russian period, including documents, letters and other vernacular texts, to signal the initial position of a word or a syllable or occasionally to mark a closed vowel (developed in North Russian dialects since the 14th century).
Other glyphs could be used in the same functions, including Monocular O and Cyrillic Omega.
The names used in literature (e.g. wide on) are just shape-based or functional descriptions.