Definition of Refugees
I think that the key to answer your question lies in understanding the precise definition of Refugees. And in particular what is the difference between a refugee and an asylum seeker.
The UNHCR states that
The terms asylum-seeker and refugee are often confused: an asylum-seeker is someone who says he or she is a refugee, but whose claim has not yet been definitively evaluated.
According to the Guardian,
A refugee is a person who has fled armed conflict or persecution and who is recognised as needing of international protection because it is too dangerous for them to return home. They are protected under international law by the 1951 refugee convention, which defines what a refugee is and outlines the basic rights afforded to them.
and for asylum seekers, they specify that
States are under international obligation to consider claims for asylum and not to immediately return asylum seekers to the countries they have fled from. The refugee convention states that they must be given access to fair and efficient asylum procedures and measures to ensure they live in dignity and safety while their claims are processed.
You might want to consider these other sources. But the simpler difference may be given by SSI as
An asylum seeker is a person who has sought protection as a refugee, but whose claim for refugee status has not yet been assessed. Every refugee has at some point been an asylum seeker.
So both are people who live their countries for fear of war, violence, etc. and political pressure. Basically anyone who believes that their own country would not grant them the free and safe life that they should be entitled to get.
The refugees are recognised internationally. And all countries who signed the UN treaties should grant them some protection wherever they go. Asylum-seekers have arrived in a country and ask that particular country to grant them asylum and allow them to stay on the ground that their original country failed to give them the correct protection. So it is a national process.
So for people living the "stable" countries, I think, as stated, it includes people who were granted asylum in another country.
Who could they be?
So, the detailed statistics are pretty incomplete (example of UK) and thus would be hard to actually give you a final answer. However, I think it falls more into the categories of people who fear they would not be treated justly, or their freedom is threaten in their home country. To give some examples
a few years ago, I heard of a German familly who "fled" to the USA and were granted asylum on the basis that the German law prevented them from home-schooling their children. School is mandatory in Germany starting from 7 yo. They felt that it was against their religious concept and philosophy.
Edward Snowden and Julian Assange have left their countries and applied for asylum in other countries and as such may appear in the statistics. In both case, they fear they would not receive a fair justice case in their own countries for political grounds.