You said your battery was "sizzling", that was probably because you had a bad cell in the battery, or maybe more than one. Battery lifetime dates are for general maintenance scheduling, if you have had trouble with your primary electrical circuit I just figured you would be better off and safer to replace it. I guess you could have the battery analyzed under a load at a service place but bear in mind if they find anything slightly abnormal their recommendation will be for you to replace it since their business it to sell you something
. As for the smaller red wire from the solenoid to the relay, if you mean from the starter solenoid then you are right. That is ACTUALLY from the relay to the solenoid. That electrical tie point routes Battery B+ from the battery positive terminal to the starter solenoid when you turn the key, there is another wire which comes from the ignition key to the solenoid which actually causes that to happen. Then also, there is an electrical tie point from the B+ to the starter motor itself and that wire should be as large as the battery cable from the positive terminal from the battery, they should both be attached to a terminal (stud or bolt, held on with a nut) on the relay. At least that is the way a basic primary circuit is wired on most cars. The B+ is always present at the starter motor and the other side of the starter motor winding is grounded when the solenoid engages the starter motor. That is the click you normally hear when the starter motor is bad, I am sure you have heard that before, the solenoid actuallly moving makes the clicking sound. If the starter motor works too then you never normally hear that because the engine starts almost immediately. I hope this helps you out and I hope I explained it well enough. Good luck.