Infinity, is clearly not finite. But there is some dissent on whether or not zero is finite. I have seen authors use "finite" to indicate the value of $0$ is excluded as well as infinity.
Is there any rule about this?
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There is no rule, and it depends on the context.
If you're worried about things being very big, then zero is an OK value to have, and you'd count it as a finite quantity.
In other situations, however, you are concerned about whether a quantity $q$ is exactly zero, or whether it is only a finite-precision approximation of it. Thus, you might say that "$q$ is finitely small", but after a while you end up simply saying "$q$ is finite" for that assertion.
It is open to discussion whether this is mathematically correct or a useful shorthand, but the fact is that people do use it and their communications are perfectly clear, so the discussion is pretty moot. Whether zero is 'finite' or not in a given paper should be obvious from the context, and if it is not then the charge against the author should be "ambiguous wording" instead of "incorrect notation".