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A true "1 wire" alternator uses an internal connection in the place of the red wire from the output to the voltage sense wire. They have only 1 wire connected to the alternator.

I used to run alternators like your setup but I now recommend adding the missing wire. Originally the missing circuit was powered with ignition power (key on) through EITHER a dash warning light OR a volt gauge. The circuit creates a small magnetic field to form in the rotor which will allow the alternator to start charging almost as soon as the rotor begins to turn. Without the circuit connected the alternator does not begin to charge until residual magnetism creates enough voltage in the alternator for the circuit to self-excite. Generally this happens around 2500-3000 alternator rpm. In many cases it's not a problem. But after accidentally running a battery down by starting the engine and letting the car idle with the lights on for a *bit* I tend to recommend the exciter wire.