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The quotes from the books are *exactly* the kind of ridiculously naive and over-simplified linguistics in introductions to logic that I criticize [in this blog article](https://www.hedonisticlearning.com/posts/the-pedagogy-of-logic-a-rant.html#silly-linguistics). Particularly for everyday natural language expressions, you will find no shortage of ways where this "translation" provides only a clumsy interpretation, is inadequate, is outright wrong, or is just nonsensical. This also has nothing to do with how mathematicians/logicians use formal logic.

Even then, your question is strange. There's no reason to expect every definition of "unless" to be even clumsily equivalent to this "translation". There's also no reason to expect the etymology of "unless" or this obsolete form have anything at all to do with this logical interpretation. This is almost certainly not the form of "unless" the authors of these books were intending, especially as the given example doesn't use "than" or "that".

Either you're asking how the natural language meaning of "if not" relates to this obsolete form of "unless", in which case this is not a mathematics question; or you're unreasonably asking how this "translation" corresponds to a meaning of "unless" that is clearly not the meaning the authors intended. It would be like asking why "not" is "translated" to logical negation but under the condition that "not" is defined as "nothing" because "not" (allegedly) derives from "nought" which means (among other definitions) "nothing".