On my starter I had the big power cable, and a smaller one the connects with a spade(?) connector on a little metal tab. I'm not even sure what it does (ground, relay, link to car computer, sensor of some sort,??)
my problem was the wire was barely holding on to the spade connector. I cleaned everything with wd-40 and fine sandpaper, cut the wire (don't cut too much mine was a tight fit), put a new connector on, reconnected the thing and sprayed a little lithium grease so it wouldn't rust as fast, and wrapped it in electrical tape (didn't have electrical shrink tube or I would've used that).
The basic starter config is a heavy duty cable coming from the positive side of the battery directly. The little wire you mentioned comes from the key by way of any interlocks (neutral/park switch on automatics or the clutch pedal switch on the manual transmission cars).
If the battery is good, and if the starter spins happily but you have an occasional no-start situation then you need to put a test meter on the little wire connection (meter = red lead on the little spade wire connection, black lead on the chassis somewhere clean) and see if you are getting 12V+ when the ignition key is in the start position.
If you have 12V+ and no starter action then the starter solenoid (looks like the smaller horizonatal can attached to the big can of the starter) is probably bad or stuck. You can tap on it lightly with a hammer while the key is in the start position and see if it comes to life. My old Beetle had this problem on hot days.
You could have a bad commutator in the starter and depending on what position the starter stopped at when last used - the brushes may be trying to power that bad spot on the commutator. Take the starter apart and test each commutator (continuity test), see if you need new brushes or skip all that and buy a rebuilt starter. I prefer to use a local shop that specializes in rebuilding starters and alternators. Just take it off and take it to them. Or - buy one from a junkyard and rebuild it. Keep your original starter and rebuild it too after the car is running and you have a spare on the shelf in case you need it.
If you don't get juice at this terminal, then go backwards and test the power out of the safety interlocks and key switch. My VW Cabrio "eats" an ignition switch every few years and this is the behavior - no start but it usually fails randomly for a few weeks before it fails completely. Since it is a 5 speed manual transmission I can roll start the car to get home. Replacement is $15 and takes 30 minutes.
You can bypass the ignition switch and interlocks temporarily by using some spare wire connected from the 12V+ side of the battery and then just touch that spade terminal briefly and the car (if the starter is good) will crank over. What the fingers and toes, don't get run over, and don't let the wire get wrapped up in the moving parts of the engine or burn through by touching the exhaust. Some starters are made so you can do this with a screwdriver carrying power from the big red cable terminal on the starter to the solenoid. It always sparks at the same time the engine tries to start so try to be prepared and don't get run over or bump your head.
If financial times are tough, you can run that wire inside the car to a horn button (momentary button) and then back out to the starter and start the car. Turn the key on, press and hold the button until the car starts. Beware because the car will still rollover (crank over) without the key turned on but it won't start.
I love working on cars and talking about working on cars. By day I'm an engineer but by night I'm spinning wrenches.
