What does the term pannier really mean? Does how the bag is hung matter?
In English, panniers originally referred to bags slung across the back of an animal (such as a horse), such that the weight hung down on either side of the animal. This keeps the weight lower so that the animal won't be as off-balance, and distributes the weight evenly.

On a bicycle, the term is used for containers that resemble that, even if it's only on one side.  Specifically, bags that hang on the side of a rack. It's also used for containers that aren't bags, such as buckets, baskets, or hard cases; though most often for bags. The term is used equally for bags on the side of a rear rack or front rack.

A pannier could attach itself to the side of a rack (typically via hooks), or you could have a single unit where two side-bags are attached to each other and held on by the attachment between them sitting on top of the rack. The attachment could be as little as a strip of cloth, or as much as a whole third bag that fits on top of the rack. Even if it's one unit, something that hangs on both sides would be pluralized as "panniers" while a single bag on one side is singular ("pannier").

If a bag is designed to fit on top of a rack without hanging down the sides, it's not a pannier. Typically those would be called a "rack bag".

A bag that hangs from the saddle (seat) is a "saddlebag". However, on a motorcycle, saddlebag and pannier both refer to containers hanging on the sides (the saddle on a motorcycle generally extends to the back). On motorcycles, "saddlebags" are commonly hard cases that wouldn't otherwise be called "bags". The huge difference between a motorcycle's "saddlebags" and a bicycle's "saddlebag" sometimes causes people to incorrectly refer to bicycle panniers as "saddlebags" even though they aren't attached to or even near the saddle (seat).