A point has been raised, and I think it's a good one. "Why are you saying we shouldn't use the Septugint for our vocabulary definitions when the lexicon you've referenced says that the LXX is by far one the most influential of their own sources?"
That's a fair question. So let me amend my previous statement with this qualifier about who "we" are. We are not everyone in the world who want to know what Koine Greek means. We are people who haven't spent their lifetimes studying the language, various papryii, documents from not only the New Testament, but also the NT psuedopigrapha, the writings of historians, philosophers, and the common people, as well as the LXX. And if you are one of those people, I would suggest that you would do well to start your search for what a word means with a reference book devoted to that purpose, instead of a translation of another document, even if the translated document had a profound influence on the development of a language.
Think about it this way. If you were going to try to found out what an English word means, would you go to the King James Bible, or would you go to the Oxford English Dictionary? Most people (the "we" here) would go to the dictionary, despite the fact that the King James Bible had a profound influence on the development of our language.
In the same way, we have no need to limit ourselves to a single source, because we have a source that discusses those words--it talks about the semantic range they have, and it discusses them in context. And that source, as I've previously mentioned, is A Greek-English Lexicon of the New Testament and Other Early Christian Literature.
By the way, as an aside, if you want to know which one you have, the easiest way to tell is the cover. Blue boards are BAG (1957), green boards are BAGD (1979), and maroon boards are BDAG (2000).
I'm still one my way to discussing the meaning of the two words in question (σῴζω and ψυχή). We'll follow that with a discussion of how people have translated James 1:21 (not interpretation yet) down through the years, and why it's significant the no one has translated the phrase "save your souls" as save your lives. And finally, we'll end with a discussion of the verse in context, and attempt to reason out it's meaning.
Hope you'll stay with me while I, er, we, make this journey together.