I am confused as to which one is technically the inset figure in the below map. I know an inset map is one which has more detail.
But in this case which one is referred to as an inset map conceptually?
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Figure 1 is the inset map.
An inset is always smaller than the primary map. Inset refers (somewhat obliquely) to a position within the page layout. It is typically completely within (ie set within = inset) the primary map, however it can be partially overlapping or completely outside but still on the same page/layout. If it were on another page/layout (ie, opposite side of a foldout map or other page in book form) it might be referred to as an inset but would more properly be called a detail map. Smaller maps on the same page can also be referred to as detail (larger scale than primary) or overview (smaller scale than primary) maps.
What is shown by the inset is irrelevant, because the term refers to the element itself - not the contents. An inset map does not necessarily show more detail. If you just switched the contents of the two figures, Figure 1 would still be an inset but at that point would be referred to as an overview map rather than a detail map. An inset at the same scale as the primary map can also be used to show areas that are separated by some distance from the area covered by the primary map; for example United States plus an inset of Alaska and Hawaii, as Random832 suggests.
Figure 2 would never be considered an inset map.
It already shows up in the Related Questions section, but just to link it a highly related question is How to produce a better inset map? Also, Farid has already linked to the Esri definition. Some other references:
I suppose FIG.1 is the inset map. FIG.2 is simply a map.
Based on ESRI GIS Dictionary,an inset map is:
A small map set within a larger map. An inset map might show a detailed part of the map at a larger scale, or the extent of the existing map drawn at a smaller scale within the context of a larger area.
According to this definition inset map is the map with smaller scale (FIG.1)