First some parallel translations for reference.
http://bible.cc/leviticus/19-16.htm
As the OP said, the version he's referencing is actually phrased differently. We could get all hung up on that, but I've always held the Bible isn't about literal wording or beliefs, it's about ideas and principles God wants us to live by, even in regards to CCW.
I rather think the point is that the verse means not so much a literal mandate to stop your neighbor from being killed by your own force, but rather to act in a manner which shows humanity and concern.
That whole idea of showing concern for other people is pretty consistent throughout the Bible actually.
http://bible.cc/john/13-34.htm
Check out the cross references at the bottom of the page.
Now as far as what it means in the context of CCW, my two cents is this, it's just my personal belief so take it FWIW. I also hold as one of my theological beliefs that the Holy Spirit can move each of us to to different answers in our lives from the Scripture, different mechanics for the same belief in other words, so my idea is not necessarily the "correct" one.
I think it's dangerous to try to take an idea, a principle if you will, and try to extrapolate an absolute set of hard and fast rules from it based on some particular wording. When you do that you "lean into your own understanding" (Proverbs Chapter 3) instead of trusting in God's guidelines.
Rather, I think this verse, and the Bible as a whole, gives us an understanding of how to try to act, and the better we understand the general ideas and principles of the Bible, the better we can apply it in context. Remember how I said I think different people might have different interpretations of the actual mechanics?
But what is that context? It's your life. Your life is unique and different from everyone else's, and the situations you encounter, while they may be very similar or even shared by many others, are ultimately unique. In regards to CCW, the context is the scenario as it unfolds.
As for me personally, I do not believe that it's my role in life to run around and try to solve or prevent every crime by force. I don't think that's realistic or the best use of my time, and it could even be dangerous. But that's not to say I would absolutely never get involved.
Yes I have competing loyalties, my family is more important to me than other people I'll admit it, but at the same time, if I were standing right behind someone who was about to kill someone else who was obviously innocent, I don't think I could live with myself if I just sat there and watched them die, and I don't think God would want me to not do anything either. Heck, it may very well be that my moral compulsion to act is God's will for that other person to live manifesting itself, who knows?
So I don't advocate taking other people's safety into your own hands, but honestly if I knew something bad was about to happen, I somehow knew who was the "Good guy" and who was the "Bad guy", and I could act and stop it, I would.
It's not likely that will ever happen, but it's not implausible either.