Is it possible to make the keyboard button toggle between wireframe and textured instead of wireframe and solid?
It sure is. Just go to Blender's preferences in the Input tab. Search by key-binding: "z". In the 3D View section you will find three "Context Toggle Values" shortcuts, one being assigned to the z key only (no modifier keys). Expand this and change the value of "SOLID" to "TEXTURED".
There is a small yet notable inconsistency in labeling that, in the UI it is listed as "Texture" yet its value name is "TEXTURED" with a D on the end.
First, note that one can already activate "Textured" mode by pressing the key combination ALTZ, and change this back to wireframe by pressing the Z key. SHFTZ activates "rendered" mode, and it, too, is canceled by pressing the Z key.
But to answer the question, yes. The key bindings can be reassigned by changing the bindings in:
User Preferences > Input > 3D View > Context toggle values.