Number of cycles in connected graphs

A few doubts regarding graph theory:

1. When considering cycles in a connected graph, we have cycles that go both clockwise and anti-clockwise. Do we consider these as two separate cycles?

2. In a complete graph such as $K_3$, how many Hamiltonian cycles exist? My guess is that it is $3!$ because each vertex connects to every other vertex and any permutation of the vertices gives you a cycle. But when I look it up online, it says there are only 2. How and why?

3. Consider simple cycles, not Hamiltonian or Eulerian. Am I right if I say that $K_3$ has $3!$ simple cycles?

A cycle is a cyclic sequence of at least 3 vertices $v_0,\ldots,v_{n-1}$ such that $v_i$ is connected to $v_{i+1 \pmod{n}}$.
The two cycles in the complete graph on $\{1,2,3\}$ according to this definition are $1,2,3$ and $1,3,2$; note $1,2,3=2,3,1=3,1,2$ and $1,3,2=3,2,1=2,1,3$.
Sometimes it makes sense to consider sequences not up to rotation, and then you indeed get $3!$ cycles in $K_3$. It all depends on your definition - use whatever is most convenient for you.