Demons

The common understanding of "demons" can refer to different types of creatures in the Nasuverse. There are the devils that became a curse created from the imagination of humans, and demon species that are demons by nature.

=Devil= Devils (悪魔) are the sixth imaginary element. They are passive creatures born and called forth from the wish of humans. They are an image that has achieved a form from the collected thoughts of humans. To materialize, they need a "form created by humans". They aren't natural devils, but things that acquire a body.

Regardless of the result, as they try to understand and remove the cause of humans's suffering, they actually are the ally of humanity. Hence they actually are the opposite of what the name implies, as they aren't creatures that punish humans. By this description, Angra Mainyu is indeed a devil who doesn't behave the way humans expect a devil to be. The description of a "red devil" is fundamentally different. This is a fact that only someone possessed by an actual devil can understand, and no one else. In fact, it cannot be confirmed until the possession becomes visible. Someone possessed will hide it as best as possible until maturation.

Demon possession
The Western understanding of a possession (悪魔憑き) is caused by a "devil". It is categorized as a curse, "something" non-human attaching itself to a human then decaying the host from the inside. Unlike the Japanese understanding of a possession by a fox spirit, the point of the curse is not known, and a devil can possess a human without any reason based on the understanding of humans.

The early symptoms of a possession is when the human starts being unreasonably offensive and coercive toward his surrounding. The symptoms of a more severe possession are when the possessing existence starts forcing its own shapeless self on the subject, affecting the external appearance of the human as well as the mind. The extent of that affection can depend on the strength of the possessing devil.

Of course, the shape of a devil is very different from that of a human, and the human can't turn back to its human state. In fact, forcing the impossible reversal will kill the possessed human.

However, there might exist some exceptions that are able to survive long enough by using the devil devouring their own soul, becoming a heresy that can endure the degeneration. However, the case of Caren Ortensia isn't an example of that, as she still is in a state where the affection is only a spiritual harm.

Otherwise, there are possessions that can diverge into poltergeists or so, interfering with their surroundings. But there is no contagion like a regular illness.

The Church has no mean to exorcise a fully matured possession, and incineration is the only way; this is the duty of the inquisitors. Of course, before exorcising the demon, one needs to find the possessed person.

=Demon= A demon (魔) is a creature that uses an unnecessary power to bend the flow of the world to their own rules. They generate a distorsion by just existing. Also related, Reality Marbles are an imitation by humans of how demons can bend the rules of nature, how demons perceive the world.

Demon species
Demon species (鬼種) are natural demons, and not humans who became demons. In the Tsukihime setting, they are creatures (oni (鬼)) that are genealogically not related to humans. They declined as civilization developped, and have now become an extinct species.

Several families have crossbred with demons long ago, creating half-demon families; the ancestors of the Tohno family have done so, hence why they have demon blood. You could say that the demon species are to the Tohno family what the True Ancestors are to the Dead Apostles.

Sources: based on the Tsukihime Laboratory entry and the dictionary resource at Jonnobi.