To move an asset type from one Category to another, open both categories and simply drag and drop, as shown below

This can also be accomplished in the menu of the selected definition by typing in either a new or existing class/category in the displayed path (highlighted in red)

Typing in the first box (highlighted in blue), will either change the class or create a new one if it doesn't exist

Typing in the second box (highlighted in green), will do the same to the category

You can also change the name of the type itself (highlighted in orange)

To create new classes and categories, you can also type in a new value (make sure to click the underlined object) when creating a new asset definition in the text input highlighted below