This is a very complex question and, you're right, there are internal definitions at play that prescribe certain kinds of information for each. Let's see if we can sort it out. Let's start with style. Literary style covers three main categories. The first is influence or literary school pertaining to classifications such as classicalism, postmodernism, metafiction, neoclassicism, Bryonism, etc. The second is the classification of overall approach to language, characters, problems and conflicts. Two examples are elevated, or formal, style for such writing as odes and epics, and a low, or common, style for such writing as Wordsworth's lyrical poems and 20th century novels. The elevated style comprises formal language and speech in narrative and dialogue; characters from the elite of society; and noble, important problems and conflicts. The low style comprises general English and colloquial language in narrative and dialogue, which even includes dialect: characters from any walk of life below the elite (there may be an elite character or two as well); and universal problems and conflicts common to all humankind, or they may even be particular to the class discussed, as in The Grapes of Wrath.
The third category of style focuses on how the writer expresses thoughts and speech within these broad categories of influence/literary school and elevation (ranging from formal to common to dialectal, etc.). Questions of literary elements that are present in all literature are encompassed in the broad categories of style discussed above (e.g., theme, which is related to problem and conflict; characters, which is relevant to elite versus common; and setting, which is related to both as well as to influence and school, e.g., a Gothic style set in a crumbling castle and a lyric style set in the country). The components of the third category of style range from tone to means of character development, including such literary techniques as tropes; allegory; imagery; mood; diction; punctuation; grammar; and pace; etc.
For your question (in which there is really some overlap of terminology because language is a component of style), in addressing the dramatic significance of language, you would look at metaphor, simile, allusion, assonance, mood, diction, figures of speech, authorial voice, tone, anything related to single words and word phrases or groups of words. In addressing the dramatic significance of style, you would look at influences or literary schools that are followed (or broken away from, as Shakespeare broke away from Aristotelian tragedy in some particulars); the kind of conflict, whether noble and elevated or common and universal to all people, etc.; and the kinds of characters, whether kings and princes or yeomen and country cobblers.
In addition, stagecraft refers to the technical aspects of staging a production; it does not refer to the artistic, or acting, aspects of play production. Some technical aspects are sets, makeup, costumes, lighting, sound, stage design, and more.