How are the elements of a dihedral group usually defined?

There is a third way, which follows from the first way, but which is just as popular: $$ D_{n}=\langle x,y\mid x^{n}=y^{2}=(xy)^{2}=1\rangle $$

From this presentation, one can deduce that the elements of $D_n$ are of the form $x^i y^j$ and so $D_n$ has order $2n$ (and is commonly also denoted by $D_{2n}$, some confusion arises...)

In practice, it is quite useful to have all these various ways to think about $D_n$ (and other groups).

The Way 2 by permutations is good only for small orders (say up to order $8$ or $10$). I have not seen its use for some specific purpose.

The Way 1 is geometric, and it is used many places, and also very useful in many places. For example, in regular $6$-gon (hexagon), we can (indirectly) see two regular $3$-gon (equilateral triangles). This implies that there are two copies of dihedral group of order $6$ in dihedral group of order $12$.

(The third way in an answer is better in some computation purposes.)