What is the right way of determining whether a digraph has a cycle or not?

Count the number of arcs: if for n vertices we have n$-1$ arcs, then there is no cycle, otherwise there is one.

Run DFS$,$ and if for the exploration of any vertex, one out$-$neighbour is a vertex which has been marked as already finished, then we found a cycle.

Run DFS$,$ derive a potential topological sorting, and determine whether it is one: the digraph is acyclic iff it is indeed a topological sorting.

Run DFS$,$ and if for the exploration of any vertex, one out$-$neighbour is a vertex which has been marked as already discovered, then we found a cycle.