I created this cheat sheet in order for me to understand and memorize all these different types of routes better. I am somewhat sure that it is fully correct, although I would appreciate it if a blessed soul would double-check it.
One last thing that I am not fully certain about: are the two "not a cycle" (at the end of the cheat sheet) really not cycles?
Edit
The final version of the cheat sheet:
Best Answer
This all sounds correct and indeed the non-examples of cycles pictured are in fact not cycles. The intuitive comparison of cycles being "like circles" is the correct one. More specifically $\color{blue}{A}EC\color{red}{A}DB\color{blue}{A}$ fails because of the red $A$ in the middle is the same vertex as what appears elsewhere. There must be only one vertex that appears more than once, that vertex may only appear exactly twice, and the positions of said vertex can only be at the start and end, not anywhere else.