The View you have to edit is the "Movies" View. Everything below that is just the Categories within the View. Those can be changed by editing the Movies View, not by right clicking on the Category and trying to change it there.
But you shouldn't really edit the standard "Movies" View, at least not until you have built a new View you like better. Then you can delete the original View, if you like. The original/standard View is a Locked View anyway, so you would have to unlock it, or force MC to overwrite it. You should create a new View based on the Movies View, and change that as you desire. The easiest way to do that is to left click on the Movies View, and drag it up to the "Video" heading, and release the mouse button. MC will then give you the option to copy the view, or Move it, or Cancel the action.
Once you have copied the View, Rename it to something that makes sense (right click, Rename). Then right click on it, select "Customize View..." to edit it. Unlock it (under Advanced in the middle section), change the "View As" on the left to "Categories" and you will see the structure you have been drilling down into. See the attached image for what that looks like.
Change the new View to your hearts content. Search the Wiki and Forum for information on how to create/edit/modify views.
But before you do all that, go into Theatre View and look at the Movies View. Go up to the menu at the top of the view, and change the selection to Genre. You will see a list of Genres on the left, and movie thumbnails on the right. Arrow down to a Genre, and click on a Genre group, or press OK on your remote control. Then you will see a display of thumbnails on all movies in that Genre, with movie names below the thumbnail. Is that what you really want to see? Maybe use Theatre View instead of Standard View then. You can also edit Theatre View Views separately under "Options>Theatre View" using the "Items to show" and "Details" sections. Again, search around for descriptions on how to use that feature.
Have fun experimenting!