Are there "lawful" enemy combatants and under who's law these ones unlawful?
Yes, there are. I'll explain in a moment. The Law in this case is International Law - the Geneva Convention, among others, is involved here.
And aren't they enemy combatants because a "coalition of forces" invaded their countries?
Yes, that is part of what makes them enemy combatants. The other part is that they chose to shoot at those invaders.
Ok, so some explanation -- there's some rules of war that the countries in power at the time put together. They include things like soldiers needing to wear a uniform with identifying marks for the country (or group in cases where you might not have an officially recognized country) in whose service they are fighting. If two of those powers went to war, they'd follow those rules (in theory), and soldiers of the other side would be lawful enemy combatants (or usually just enemy combatants, contrasted against enemy civilians).
If some of those soldiers stripped off their uniforms and did stuff against those rules, they could be disavowed by the other country -- they were out of uniform and therefore they were unlawful enemy combatants. The special rules regarding the treatment of Prisoners of War wouldn't apply. They could be held after the cessation of hostilities, for example, and could be tried by the country that captured them for their crimes rather than those acts (such as mass-homicide and such) being considered acts of war and therefore somehow perfectly acceptable.
So if these insurgent groups wore a uniform of some sort, and followed a normal command structure, and didn't hide in civilian populations, they could be lawful enemy combatants. They'd also be a lot easier to eliminate, which is why they don't do that. However, because they aren't playing by the Big Powers rules, that means the Big Powers don't technically need to follow those rules either. I still think we should, but that's a separate discussion.
That should hopefully help you understand where the term comes from, and why it gets used in reference to actions like this.