Is there a flaw in this Wikipedia proof of cycle property of Minimum Spanning Tree?
First off to answer your main question: there is no flaw in the proof. 

The point where you are not following the reasoning of the proof is that:


E belongs to a cycle in the graph and to the MST. 
Not every other edge of the cycle are in the MST. 


If you look at your drawing, then either one of the edges towards the rest of the MST is redundant. Because not all edges of the cycle are in the graph results in that if you remove E you get two subtrees.

To answer your question about why the proof begins with assuming that e belongs to the MST: Proof by contradiction starts with assuming the opposite of what you are trying to proof. Then you reason with that assumption until you run into a contradiction.
A contradiction means that your assumption must be false. If you assumption is false, then the opposite of your assumption is true.

If you just look for images and MST examples there are enough examples with a cycle. Try to follow the reasoning of the proof and you will see why it is sound.