But regardless, not every agreement is ...
> legally-binding
That term has a different, and as I understand, more nebulous meaning under international law?
Well, even the US agrees that "international agreements" are "treaties" in the international law sense, despite not being "treaties" in the US domestic law sense.
> Any idea where I can read a bit more about the fundamentals
A lot of what I know about this topic I learned from reading the Third Restatement of the Foreign Relations Law of the United States – https://www.ali.org/publications/show/foreign-relations-law-...
Unfortunately it isn't easy to get your hands on. You can buy a hardcopy for US$173 plus shipping. Or you can do what I did, and read it for free in a university library. (It is also included in Westlaw subscriptions, but unless you already have access to one, buying the hardcopy would probably be cheaper.)
The international law on this topic is mostly contained in the Vienna Convention of the Law of Treaties of 1969 – https://legal.un.org/ilc/texts/instruments/english/conventio...
> That term has a different, and as I understand, more nebulous meaning under international law?
The most common way to make an international agreement not legally binding is to put a clause in there explicitly stating that it isn't legally binding. When the agreement explicitly states it isn't legally binding, then it clearcut isn't.
If an agreement is in the usual written form of a finalised formal agreement, it is generally going to be assumed to be legally binding unless it explicitly states it isn't.
Generally speaking, to be binding under international law, the parties have to have "international legal personality". That basically means the parties must be the national governments of sovereign states, or international organizations established by treaty. An agreement involving private corporations, private individuals, subnational governments, non-governmental organisations, etc, generally isn't going to be legally binding under international law, even if it also includes national governments among its parties. Occasionally, dependent territories are granted power by the national government to sign legally binding international treaties on certain topics (such as Hong Kong and Macau), but that is an exception to the general rule.
There are grey areas which lawyers and scholars will debate, but it rarely turns into a live issue in practice.
Out of curiosity, what makes you so interested? I'm interested, but I'm usually alone in that.
Only in the US (and possibly a handful of other countries which copy the US approach). Under international law, all legally-binding international agreements are treaties. What the US calls "international agreements" are treaties from the non-US point of view.