So, unfortunately, this is complicated, because as often as not people mean something different from the original sense, and you kind of have to guess from context.
The term means original equipment manufacturer — a company that makes components. As you probably know, nothing complicated these days is built all in one factory. And, in fact, not all of the parts come from one company. Cars, cameras, computers — you name it, really… for any of these things, parts are sourced from all around the world from many different companies specialized in certain areas.
So, an OEM part is a part made by one of these supplier companies. It's not usually meant to be sold directly to consumers, but to the "final product" company who will put their name on it.
However, often these components are available directly (or through gray-market sales channels). This is where the confusion comes in. For example, an "OEM battery" may be any of these things:
- The official battery made by the contracted supplier with the final product branding and everything: that is, the "legit" thing.
- That exact same battery made by that same supplier, but without the branding on it. (In computers, this might be called a "white box" version, but this isn't particularly common in cameras.)
- A similar battery made by a different supplier that is not actually contracted for the brand-labeled part.
Because camera makers (or computer makers, or car manufacturers, or whatever) shop around for the best deals for suppliers, it may even be that an OEM in the third case was at one point the official supplier but is not anymore.
To make things even more confusing, there's a totally common different use of this term, where people use it to mean "labeled with the big brand" — the "original" camera brand instead of third-party. I'm pretty sure every use of "OEM" in the links in the question actually means it this way. This is very arguably wrong — but… it's probably more prevalent than "correct" use, so, well, look out for that.
- This term is confusing and I recommend using something more clear when writing or speaking.
- If you see this label on a product you're about to buy... buyer beware. Make sure you know exactly what the seller means (and don't trust any pictures).
OEM, which stands for original equipment manufacturer, is one of those terms for which the meaning can vary greatly depending upon the context in which it is used.
It can mean one thing in the automotive manufacturing community (an OEM is a company that supplies the manufacturer with parts for whatever it is that the manufacturer produces for retail sale), another in the computer manufacturing environment (where OEM usually means a part that is made by a supplier to a manufacturer that is made to the same specifications as those sold to the manufacturer - whether or not the part was sold through the retail network of the "brand" for which it is made), and yet another in the camera community.
When most people in the field of photography use the term OEM, they generally mean the item or part sold by the "official" brand. If someone says a part is "Canon OEM" that means the part is one actually branded and sold by Canon through their authorized retailer network, even if it was actually manufactured by one of Canon's suppliers.
Conversely, if someone refers to an accessory, lens, flash, battery, etc. as being non-OEM they generally mean it is not sold through the "brand's" supply chain and is not identified as an official product of that brand. Another term many people in the photography community use is third party. Third party and non-OEM are essentially interchangeable.
Take batteries, for instance. Canon does not make their own rechargeable Li-Ion batteries. Apart from counterfeits, Li-Ion batteries that have the Canon Brand on them are made by a supplier and sold to Canon, who then sells them through authorized Canon retail channels. This is what the vast majority of people mean when they say they want or have an "OEM battery" for their Canon camera.
Batteries made to the same specifications by the same manufacturer who supplies Canon with their OEM batteries but sold through other retail channels would not be branded "Canon" and would not be considered "OEM batteries", even though they are identical other than the name printed/stamped on them.