The smaller wire is the exciter and the large wire is the output wire.
Alternators and generators make electricity by rotating spools of wire through a magnetic field. In the case of a generator, the magnets are usually permanent (metal or ceramic). In an alternator, it's an electromagnet made of copper coils. The wire that powers the electromagnet 'excites' the coils, providing the magnetism to make power.
One advantage of an alternator is that if you don't need electricity from it (battery is fully charged) then the electromagnet can be 'turned off' and at that point you just have the bearing friction from the alternator, where a generator will try to supply power anytime it rotates.
The alternator can vary its 'exciter' power infinitely (well, between 'off' and 'full'), which is why you will notice that when you add electrical load, the alternator's 'whine' changes pitch - the electromagnet is pulling harder and the coils are moving through more magnetism, so they make more power. This also means the alternator can make more power without raising engine RPM.
One downside is that an alternator needs voltage in order to 'start', and needs a battery present in order to regulate itself. Generators don't have to have a battery to work.
Alternators also have diodes, since they make AC power that needs to be converted to DC power to run the batteries.
Power will be present on the small wire when the ignition is on.
One check you can do is to set your multimeter to AC and check the output (multimeter red to large wire and black to ground) of each alternator. The results should be close to each other and very low (a couple volts or less). High AC voltage on one alternator means a failing diode in that alternator. NAPA does carry replacement diode packs and regulators, but replacing the whole alternator may be worth it to you since that means new bearings, too.
A full test could be done with a portable tester if it's done right instead of dismounting it, but the second alternator may confuse the readings, which would be why the TM says pull it and put it on the charging stand so you can see the results independently.
