Finding Connected Components Dependent on Order?
There's a few things we need to clear up.

First, the graph you show is directed. Are we talking strongly-connected, weakly connected, bilaterally-connected, etc. What you mean by connected here matters.

Secondly, in such an algorithm, usually you keep searching until each vertex in the graph has been assigned a component.

So, if you use the algorithm(s) properly, and keep searching until all vertices have been classified, the output will not depend on what vertex you start at. Each vertex belongs to exactly one connected component (whichever variety you choose), and a correct algorithm will assign it as such, regardless of the start vertex.

However, if you start at a vertex, you might not find all vertices just in the search starting from that vertex, which is why you need to repeat your search until you have seen all vertices.