Are there mathematical contexts where "finite" implicitly means "nonzero?"
The problem is that physicists are more influential than mathematicians. They routinely consider zero to be a nonfinite quantity, probably because they are thinking logarithmically. If you hang around physicists, you will hear expressions like “very small but finite”.
But the concept of infinity is a mathematical one, not physical, and certainly mathematicians rule in this matter: zero is most certainly finite.
There is really no point in insisting that a definition in a dictionary has any implication on the mathematical meaning of the word. Germs have nothing to do with real world germs, and cardinals have absolutely nothing to do with the catholic church. Normal spaces are not those which are not irrational, and real numbers might not really exist (e.g. if the universe is finite).
In mathematics $0$ is a finite number, and $\varnothing$ is a finite set.
One might argue whether or not $0$ is a natural number, and that might be open to debate between mathematicians. If you define "finite" as having the cardinality of a natural number, and $0$ not to be a natural number, then indeed the empty set is "not finite", and so $0$ is not finite. But it seems like a very artificial thing to say. If you won't include $0$ in the natural numbers, then you'd define finite as empty or having the cardinality as a natural number.
The lesson here is to stick to the definitions, as best as the conventions you follow allow you. Don't get swayed by the natural language meaning of the word.
It seems to be not unheard-of to speak of small but finite quantities in applied mathematical fields. At least in this context, "finite" is obviously meant to mean "nonzero", or perhaps "not infinitesimal".
Additionally, in abstract algebra, it is not unusual to speak about rings/fields of "finite characteristic" to mean one whose characteristic is not zero. (Generally in ring theory, zero often behaves like the limit of ever larger elements, at least intuitively speaking, so it takes on some properties of "infinity").
Note that the dictionary definition you link to doesn't claim that 0 cannot be finite, period. It lists three different mathematical usages of the word, not three conditions that all have to be satisfied before you can (according to the dictionary) call something finite. Only one of these possible meanings is "not zero".