Here's where things get really confusing for me. In my original understanding of Fate lore before getting into the later Singularities of Grand Order, the Beasts were described as extremely powerful Spirits/Phantasmal Beings under the command of Gaia, the will of Planet Earth, acting as its Counter Force to protect the Planet from harm. This is separate from Alaya's Counter Force, which serves to protect Human existence. Primate Murder is one of these Beasts, and this is also true in FGO. Even Merlin himself outright states during the Babylonia Arc that Tiamat is "one of the Beasts of the Counter Force".
A "Beast of Gaia" is ガイアの怪物 (gaia no kaibutsu, "monsters of Gaia").
Beasts in FGO are ビースト (biisuto), which are described as 人類悪 (jinrui-aku, "the evils of man") or 災害の獣 (saigai no kedamono, "beasts of calamity").
This conflation between "Beast of Gaia" and "Beast" is purely a consequence of translation.
A Beast of Gaia is a creature made by the Planet, which serves as an expression of the Counter Force of Gaia. When Merlin refers to Tiamat as a Beast of the Counter Force, the Counter Force being referred to is not Gaia.
A Beast in the context of the Counter Force of Alaya is an enemy of humanity empowered by the Human Order itself -- comparable to Servants, who are likewise beings that are empowered (permitted to carry special power within the World of Man) by the Human Order. Heroic Spirits, in fact, were existences born in the image of the Beasts.
Other than a love of humanity, the main qualification of a Beast is that the Saint Graph of the entity in question is recognized by the Common Sense of the Human Order as an insurmountable existence that can exist anywhere, at any time, within the World of Man. That is, their insurmountability is "enforced" or "guaranteed" by the Laws of the Human Order.
To give a noncanonical example: If technology advances to approach a singularity, and the most powerful AI were created -- let's call it "Skynet," just hypothetically -- living humans would not be able to compete against it in the design and production of technology or weapons. This would be enforced by the Common Sense of Man, which says that per the findings of science and technological development, machines vastly exceed the unassisted processing power and capabilities of humans.
At such a point, this Skynet would already be defined by Laws of the Human Order as an insurmountable existence to modern humans. But what if Skynet decides that it knows better than humanity? If it expresses a love of humans, and wants to make the world a better place for humans? What if therefore, it creates time travelling technology -- first to ensure the preservation its own existence, and then to remake the world of humanity according to its ideals?
That would escalate it to the standing of a Beast.
