you are correct about the thick red wire. it is attached to the starter. if the starter on your machine isn't engageing then the bendix has gone out on your starter. you do not have to replace the entire starter though. you can simply take the starter off and take it to the lawnmower repair shop and have the bendix replaced. or you could do the job yourself: before you do anything" though you need to remove the red cable( the positive cable from the battery) this will assure that you don't short out the battery. If you indeed want to do this work yourself you will need to have at least some mechanical ability. But based on your prior statment about the cylindrical shaped object under the hood of your machine. If I were you I would take the machine to the repair shop. ( Sorry but in my opinion you do not have the mechanical ability to undertake such a task) hope I have been of help to you:
Mickey-G, August 2008