Below is a rough outline of a mine layout, showing the difference between a mine shaft and an adit and a winze and a level.
What most people call a "mineshaft" is, in fact, an adit - a horizontal or near horizontal access into a mine from the outside world. A shaft is a vertical or near vertical access into a mine.
Within a mine, a horizontal passage that does not extend to the outside world is called a level (an adit is a level that extends to the outside world) and a vertical passage that does not extend to the outside world is called a winze (if it opened out into the outside world, it would be a shaft).