What exactly is apple cider?
What is apple cider?
In earlier times, it just meant &quot;apple juice&quot;. Nobody had different words for the non-alcoholic and the alcoholic variety, because the freshly pressed (and sometimes cooked) juice fermented on its own when stored unrefrigerated.
In the days of refrigeration, there are different products made by pressing of apples, and stabilized to keep their properties when refrigerated. It just so happens that linguistically, the name &quot;cider&quot; has stuck for a non-fermented variant in the US and for a fermented variant outside. Besides, there are companies who have been using modern technology to approximate the fermented version without following the exact process, this is how products like the Apple bandit come into being. (There are also companies who approximate the non-alcoholic variety, but I think they market it simply as apple juice).
So the word has no single &quot;exact&quot; meaning. All three types are a cider.
Which one is better to use?
The answer may sound disappointing, but it is best to use the type meant by the recipe author. Ideally, the author would have known of the linguistic problem, and specified what they mean. Since the ideal case rarely happens in reality, you have to take your best guess based on what you know of the recipe source. That is, if it is an American recipe, use unfiltered apple juice, if it is European, use the alcoholic kind. Also, especially for older recipes, or ones which require much manipulation (such as your reducing step), it is preferable to use traditional-process cider. The modern style cider  like Apple bandit is fine-tuned to taste good when drunk straight from the bottle, but it might behave very different from old time cider when heated.
In the end, if you have no idea where the recipe came from or who the author is, you can just start with whichever type is easiest for you to use, and see if you like the result. If yes, stay with it. If not, try with a different type.