One of the small terminals should also have battery voltage (maybe a little less) The other small terminal should have some voltage (whatever flows through the bulb) Basically MOST internally regulated alternators should have some positive voltage going to all their terminals with the ignition on....
This could be an ignition coil, spark plug or spark plug wire type of problem and the engine is “missing”. To rule out a head gasket issues, top off with water and get a cooling system pressure test to determine if you do have an internal or external coolant leak. A blown head gas...