According to the definition you quote, “unless” gives an exception to a preceding negative statement. An exception to a statement is a condition in which the statement does not apply. Therefore the statement does apply only if the condition is *not* fulfilled (because otherwise the exception applies, and therefore the statement does not). Therefore, as far as logic is concerned, “unless” translates into “if not”.
In describing an exception, the “unless” phrase *also* suggests that the condition is generally not fulfilled, but that part is irrelevant from a logical point of view (the logic only cares about the truth/falsehood of statements, not about the probability that they will be found to be true).