To use the delete vertex control (should this be called remove vertex ?), you have to click on a feature to select it, then select a vertex and then press the backspace key on your keyboard to remove the selected vertex.
Unforunately appears to fail both as the online demo, and in this local context??? Locally even the WKT POLYGON is not drawn. Also locally, the function OpenLayers.Control.DeleteVertex() does not appear to exist. Is this a later/earlier version of OpenLayers.js? No, the online uses a OpenLayers.js which generate an 'include' for MANY js files, including "OpenLayers/Control/DeleteVertex.js", which is obviously NOT included in the 'single' model OpenLayers.js file.