I suppose that any national sovereignty (government) would have an expectation that as their citizens travel abroad, they would not do anything that would cause an international stir. While the government may have travel advisories regarding safety and local customs, their citizens are pretty much left on their own. They are not “official” representatives of their country. (Regarding advisories about local customs, decades ago, working as a shipping clerk, I read a list of things that Americans would think nothing of but in other countries are very offensive. One of those things was the hand gesture for “ok.” I was shocked to learn how that seemingly innocent gesture is interpreted in other countries.)
However, official representatives from a nation are held to a higher level of responsibility. Due to the fact that officials (such as ambassadors) are in a foreign country for the expressed purpose of representing their respective government, they are expected to “walk the walk, and talk the talk.” Everything from how they talk and dress are outlined. If they are involved in sensitive negotiations, it is expected that they have enough professionalism that, even when under pressure, they won’t “fly off the handle” and have an angry outburst complete with obscenities and intentionally offensive gestures. But what in the world does that have to do with the title a “Light of the Nations”? Everything.
Some might view the Law of Moses (which is really inaccurately named because it was not his but rather God’s law), as being petty, exacting, and without logic. Jesus later helped the Israelites living in his day to summarize the key points of the Law. But what of all those laws regarding physical, spiritual, and moral cleanness? Isaiah 49:6 might shed some light on that. That passage has two components. The first is that the nation of Israel was considered (by God) to be in a special relationship with him that no other peoples were. Likewise, ambassadors are privileged with a station of service that carries serious responsibilities.
The second component is the most important. Besides ambassadors being in an honored position (their relationship with their own government), ambassadors also have a public segment to their responsibilities – their relationship to foreign nationals. This second component of the passage in Isaiah is exactly what Jehovah was telling the nation – They are a “light to the world,” a shining example of what it means to truly be “God’s people,” with the eventuality that those who were not God’s people could be reached with the message of salvation. (Although there are other examples, one of the most interesting was the case of the Gibeonites and how they recognized that the true God really was the one that the Israelites worshiped.) So it was imperative that the nation of Israel conduct itself in such a way that people of the nations could see the higher standard of living they had. Cleanness and purity were concrete concepts in the Israelites minds and life.
When Jesus came on the scene, he likewise informed his followers that they were to consider themselves ambassadors to the nations. First to Israel, then Samaria, then to the most distant parts of the earth (as men of the nations were welcomed into Christianity.) (See additional info here.) In order for true Christians to be that light, they would obviously have to do more than merely live a good life. How would observers differentiate them from mere philanthropists if it were not for the outreach (preaching) work? But this doesn’t discount that we should live with high moral standards. Indeed, if Christians act just as poorly as non-Christians, that would be a huge reproach on “the King of kings, and Lord of lords.”