It's just convention.
The TEX system is usually written as TEX followed by a number, like "TEX40." It's usually only used to describe thread rather than fabric. This might be different in Europe, I don't know.
Denier also really only applies to thread, but it has become common to use it to describe fabric woven from a certain size thread.
There are at least two other systems for indicating the size or weight of thread. One is an older system, used for cotton thread, and the other is a "commercial" system, which is apparently the denier divided by 10, with a preceding letter indicating the material. "V46" is 460d polyester, for example. A "Z" apparently indicates nylon.
To add to the confusion, some manufactuurers use their own numbers to identify thread. For instance, Gutermann "Mara 70" thread is a TEX40 thread.

As far as coatings, I think, for clarity, it's best to differentiate "coatings" and "surface treatments."
The term "coatings" generally refers to a factory-applied permanent waterproof coating, usually polyurethane or silicone in tent fabric.
"Surface treatment" usually means a chemical treatment applied to the other side of the fabric, (not visible) to increase water repellence or UV resistance, fairly durable, but not permanent, and renewable by the end user.

Regarding sources, Wikipedia is pretty good, but mistakes exist. A lot of what's on the internet is just stuff that has been copied from another site, and so both good and bad information proliferate.

I've never found a really comprehensive source for information on thread and fabrics. I started making some of my own backpacking gear in the 1970s, and have just picked up knowledge over the years. I don't consider myself an expert, but I like to think I have a reasonably good "crap detector." I suspect the only real experts are those who have worked in a related industry for many years.