We once featured an Euler diagram that explained the British Isles, the United Kingdom, and Great Britain. This video explains all that clearly but quickly, then goes on to explain the British Empire, the Crown Colonies, Crown Dependencies, and other terms that confuse Americans and others who don't deal with such geographical concepts every day. If this goes too fast for you, the script is available from C. G. P. Grey. http://blog.cgpgrey.com/the-difference-between-the-united-kingdom-great-britain-england-and-a-whole-lot-more/ -via reddit
On the face of it, the video seems to present a really clear, entertaining and well-researched review of the array of names associated with the British Isles. It is true that the country is often referred to as the Republic of Ireland and ordinarily that's fine by me, but when a video like this claims to present the official picture an error like this really calls into question the accuracy of the rest of the content which is a real pity ( it's one of the first things in the Wikipedia article which casts real doubts on to the creator's sources).
Which leads me onto the other pity that you automatically assumed that I was talking about this in terms of a North/South issue. I would identify myself as Irish and from Ireland and like your Irish friends would use these terms collectively. As I gather from your experience of Irish people, none of them are too shy to let you know when and where they individually make their distinction and personally, though not ordinarily, I make mine too (and that distinction usually is that I'm not from Dublin, though half of me is).
So I guess this really about me being a pendant rather than a political aggressor (though at times I'm sure people may argue which is worse!).
I know several Irish nationals who refer to their country as "the republic" (as if there were only one) and Ulstermen who talk about it as "the south". Why? Because Ireland (or Eire) to many still exists as the whole island and the Irish are the people who populate that island. If these people talk about Ireland or the Irish they are not differentiating between north and south. And that's why names like the republic of Ireland are in common use.
You may have an Irish name, but you do not have an Irish heart if you use the name "Ireland" only to refer to the republic.