How might one define magic logically?

What a word is defined as, on its own, is beyond the scope of pure logic. It's to do with facts. For instance, if I claim that X is defined as Y, when it is actually not, this would just be factually incorrect - nothing to do with logic. So the definition of magic would be something you'd check a reliable dictionary for.

E.g. "Magic is defined as the scientific application of one's willpower" - is a false definition. Not fallacious, just false.

How definitions are used can be analysed logically when used as part of an argument. For example, the definist fallacy. If someone defined magic in a misleading way to push a certain viewpoint, it would fall under this category.

E.g. "Let's define magic as unseen forces. You believe in gravity right? That's unseen, right? Therefore, you must believe in magic!" - this is fallacious since the real definition wouldn't allow the person to make this argument.

Magical thinking has nothing to do with the above sense of magic - it refers to an irrational way of thinking where a person makes causal connections between things that are not causally related, and they do this solely based on superstition (e.g. believing that numbers can be unlucky).

As a purist one first would need to differentiate between magic or prestidigitation, or magic as in some sort of supernatural display, or Magick as defined by Aleister Crowley, as the Science and Art of causing Change to occur in conformity with Will.

Magical thinking is not really a class of magic, but simply a willingness to ascribe power to one's instinctual cognitive processes over intelligent and rational reasoning and problem solving.