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:

1. The official battery made by the contracted supplier with the final product branding and everything: that is, the "legit" thing.
2. 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.)
3. 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.

Overall:

1. This term is confusing and I recommend using something more clear when writing or speaking.
2. 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).