In the modern era an episode primarily refers to the discrete element of some form of media such as a radio play, television show, or podcast. Since the beginning of broadcast media it has been necessary to fit content to fairly tight scheduling in order to avoid dead air and assure shows occurred at their appointed times. This and the simple beginning to end structure of time and narrative form the basis of the episode. A show comes on, runs, is subdivided by commercial breaks, and ends in the span of its time slot. While it's possible to generate any sort of narrative under these constraints the obvious and natural format is neatly separated plots which are largely independent of each other despite sharing characters and setting. This fits nicely with the Webster definition; explaining the choice of nomenclature.
Episodes can be contrasted with chapters. Books are typically divided into chapters. This is not a necessity but beyond a certain size it becomes a boon just for keeping track of where one is in a tome. That said chapters can vary wildly in length according to the sense or caprice of authors and editors and end at satisfying or suspenseful moments as the tone requires. In contrast, broadcasted episodes are not lined up front to back between book covers but are shuffled into the line up of various other programs. For this reason most shows have episodes which function independently of one another and can be watched in nearly any order without significant loss in comprehensibility. Shows displaying this quality are said to be episodic.
Following the advent of digital distribution the constraints of the time slot have dropped away. Consequentially, the average episodicity of a given series is also falling off. It is now practical to have plots that are continuous from one episode to the next allowing for more intricate stories. In this sense episodes are diverging from the original implication and may be becoming vestigial. It strikes me as unlikely that media is going to be released in eight hour blocks but there is significantly less preventing it in the age of streaming.
Outside of fictional narratives the term is used for most forms of media. Talk shows and educational podcasts are both typically described as having episodes. Oddly, movie franchises (with the exception of Star Wars) use the term installments and video essays tend to just say parts. The distinction between an episode and any other descriptor seems mostly arbitrary but that's the evolution of language for you. It's likely that media that is simultaneously very discrete yet appurtenant will continue to use the term for as long as English is spoken.
IRON NODER XVII: ALL'S FERROUS IN LOVE AND NODING