$D_{2n}$ usually denotes the set of symmetries of a regular $n$-gon, where by "symmetry" we mean a movement of the $n$-gon in 3-space that gets us back to occupying the original position. However, I'm curious why we define it to be in 3-space. Obviously, from a practical point of view, this has a physical/geometric interpretation, but I'm wondering if that's the main reason why we define it that way.
For example, if we restrict our space to 2-space, then we cannot "flip the $n$-gon over", so the set of symmetries doesn't include reflections; only rotations. Then the order of the group is only $n$, not $2n$. And if we are considering, say, the symmetries of 3D objects, is our space still restricted to 3-space? Or would it make any sense to speak of moving the 3D object around in 4-space (or higher), just like we did with our 2D $n$-gons in 3-space? Are these considerations even useful? Do we care about spaces other than 3-space (or 2-space)?
The dihedral group is a fixed group that does not change based on context, except given the order. If you restrict the symmetries of the regular polygon that are allowed, you will obtain a subgroup of the dihedral group, as the dihedral group is the group of all possible symmetries of the regular polygon.
I've never heard of defining the dihedral group as being a group of symmetries in $3$-space, and it's not necessary to extend to $3$-space unless you are only allowing rotations. Reflections are also symmetries of the polygon, and if you combine the rotations and the reflections in $2$-space you get the complete dihedral group. In any case, the dihedral group is intrinsic, meaning you can define it without reference to any ambient space. Namely, if you consider the $n$-gon as a metric space in its own right, the dihedral group is the group of isometries of this metric space.