OK, I'm not familiar with this setup but here's what I can infer from your pics along with an explanation of why I think it is like it is:
Traditional starter motors (before permanent-magnet types) were series-wound motors, as these give the best torque characteristics. Most dynamos were shunt-wound, as this gives easy control of voltage using an external voltage regulator that interrupts the shunt field circuit. Some earlier dynamos used a 'third brush' to power the shunt field, exploiting an obscure characteristic of the dynamo to make it crudely self-regulating without an external regulator.
The conventional 'modern' dynastart incorporates both a series field for starting (terminal 30) and a shunt field for generating (terminal DF) to get reasonable performance of both functions from one machine. When the starter button is pressed, current flows from the battery through the series field to the armature, making a series motor. Once the engine is running, the voltage generated by the armature energises the shunt field and closes the cutout relay, delivering output back to the battery without the series field in circuit. The cutout relay is necessary to disconnect it from the battery when the engine is not driving it fast enough to generate; as soon as its voltage falls below that of the battery it will motor instead and try to drive the engine.
A shunt-wound dynamo will run as a motor when its armature is connected to the battery, just not as 'torquey' a motor is if it had the benefit of the series field. Thus, it is possible for a dynastart to have just shunt field and armature leads, as I think the units you mention in other Panhards have. They would run as a shunt motor or shunt generator according to speed, motoring when the engine is stopped and generating when running fast enough to overcome the battery voltage. Either a vibratory regulator or a manually controlled resistance could be inserted in the shunt field circuit to limit / adjust the charge rate.
The control box depicted in the circuit diagram appears to contain a normal cutout relay. When the engine is stopped, this is open and the dynastart is disconnected from the battery. Pressing the starter button bypasses it to run the dynastart as a motor. Once the button is released, it is again disconnected until it builds up enough voltage as a generator to energise the cutout, which closes and connects it to the battery. The photo of your 'control box' shows a 1960s semiconductor diode instead. This can fulfil the function of the cutout, by preventing the battery discharging into the dynastart when the engine is stopped or running too slow to generate a voltage in excess of the battery voltage. Once it rises by a few tenths of a volt above battery, charging current can flow through the diode. I presume the original electromechanical cutout relay failed and this was the alternative solution used, which should work OK.
Now we come to the question of regulation. The brass link across the two small terminals of your machine suggests that its shunt field runs at full strength all the time, there being no regulator or adjustable resistance in circuit. This would tend to cause it to overcharge or become overloaded at high engine speeds, if it were just a normal dynamo with its output connected directly to the battery. Perhaps it is partially self-regulating - a clear picture of the brushes that shows their relative positions and connections would be very helpful in clarifying this. An alternative is that the output, when connected to the battery via the cutout rather than via the starter switch, has a resistance inserted to limit the charge current even at high generated voltage. Tracing the wiring would clarify this point.
You have a charging problem. To troubleshoot it, first we need to prove beyond doubt what the intended means of regulation was. Having no other information here, to help you further I would need some questions answered by tracing wiring and taking readings with a multimeter. My first would be:
Is the diode (shown in the second pic) connected directly in parallel with the starter button, without any charge-rate limiting resistance in series?
Does the dynastart have the same number of brushes as field poles, spaced at equal angles around the commutator?
If you remove the link across its small terminals, can you confirm the resistance from each of them to the case, the output terminal and the brush-holders (you will need to slip something under each brush to isolate them from the commutator for this test).
How much effort is needed to refit the machine to the engine and run it up for tests?