The wyvern is not a folk creature that heralds happened to draw. It is a term in a technical vocabulary. Blazon, the language in which coats of arms are described, has to be exact enough that two artists working from the same words produce the same shield, so it distinguishes creatures by countable features. A dragon has four legs. A wyvern has two, with wings, and a barbed tail. That is the whole of the difference and it is a difference of description rather than of nature.
English practice hardened the distinction more than Continental practice did. In much European heraldry the two-legged form is simply what a dragon looks like, and the separate word is unnecessary. This is why medieval texts that mention dragons cannot be sorted into wyverns and dragons: the category did not exist when they were written and was applied afterwards.
The device has had a long civic life. A winged two-legged beast appears as a standard associated with Wessex and is used to this day by regiments, councils and clubs across the English west and midlands, and it is on the arms of several cities. Its modern visibility in games and fantasy fiction, where it is usually a smaller and less intelligent dragon, is drawn from heraldic illustration rather than from any narrative tradition, because there is no narrative tradition to draw from.