Much is known about the behavior of the (undirected) random graph $G(n, p)$ in the Erdős–Rényi model, including the following:
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If $np < 1$, then a graph in $G(n, p)$ will almost surely have no connected components of size larger than $O(\log(n))$.
If $np = 1$, then a graph in $G(n, p)$ will almost surely have a largest component whose size is of order $n^{2/3}$.
If $np → c > 1$, where $c$ is a constant, then a graph in $G(n, p)$ will almost surely have a unique giant component containing a positive fraction of the vertices. No other component will contain more than $O(\log(n))$ vertices.
If $p<\tfrac{(1-\epsilon)\ln n}{n}$, then a graph in $G(n, p)$ will almost surely contain isolated vertices, and thus be disconnected.
If $p>\tfrac{(1+\epsilon) \ln n}{n}$, then a graph in $G(n, p)$ will almost surely be connected.
Thus $\tfrac{\ln n}{n}$ is a sharp threshold for the connectedness of $G(n, p)$.
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The directed case is studied here.
A loop is commonly defined as an edge (or directed edge in the case of a digraph) with both ends as the same vertex. (For example from $a$ to itself). Although loops are cycles, not all cycles are loops. In fact, none of the above digraphs have any loops.
Cycles are usually defined as closed walks which do not repeat edges or vertices except for the starting and ending vertex. This definition usually allows for cycles of length one (loops) and cycles of length two (parallel edges).
Note that cycles (and walks) do not make any reference to the orientation of the edges in question. Directed cycles (and directed walks) may only travel along the "forward" direction of the edges. In particular, that implies that $G_3$ pictured above has a third cycle, $(\color{blue}{(a,b)},(b,c),(c,a))$ where the $\color{blue}{(a,b)}$ refers instead to the edge pointing from $b$ to $a$.
Technically, all of the graphs above except for $G_2$ are directed multigraphs since in each you have parallel edges. Although in simple graphs (graphs with no loops or parallel edges) all cycles will have length at least $3$, a cycle in a multigraph can be of shorter length. Usually in multigraphs, we prefer to give edges specific labels so we may refer to them without ambiguity.
As for being strongly connected, yes all of them are and your definition is correct.
Your additional question, "what is the difference between a cycle and a connected component"
The above graph contains a cycle (though not a directed cycle) yet is not strongly connected.
One can prove that if a directed multigraph is strongly connected then it contains a cycle (take a directed walk from a vertex $v$ to $u$, then a directed walk from $u$ to $v$. Any closed walk contains a cycle).
One can also show that if you have a directed cycle, it will be a part of a strongly connected component (though it will not necessarily be the whole component, nor will the entire graph necessarily be strongly connected).
Best Answer
For a strongly connected graph (or component), we desire for any vertices $x,y$ in the graph (component) there exists a directed path from $x$ to $y$.
Remember that a directed path from $v_0$ to $v_k$ is a subgraph $P=(V,E)$ where $V=\{v_0,v_1,v_2,\dots,v_k\}$ and $E=\{v_0v_1,v_1v_2,v_2v_3,\dots,v_{k-1}v_k\}$ and each $v_i$ is distinct. (Here, the notation $v_1v_2$ refers to the directed edge from $v_1$ to $v_2$).
Note that $(\{a\},\emptyset)$ is a perfectly valid path of length zero with both endpoints $a$. Your misconception seems to come with the idea that a path must have length at least one but that is not the case. As there does not exist a directed path from $B$ to $A$ and since singleton vertices count as strongly connected under this definition, your graph does indeed decompose into the two strongly connected components $\{a\}$ and $\{b\}$.