That does sound rather confusing. Generally a “built in diode” is referring to a EMF kickback suppression device. When the power to a coil in a contactor is removed, the residual magnetic field collapses very rapidly producing a potentially destructive voltage to your solid state control device. Generally the diode will also have a resistor connected to it to protect the diode should someone connect it improperly. Without the resistor, the diode will be destroyed when connected improperly.
The resistor also increases the risk of damage to the solid state control device. That being said, a diode in the output leg of the control device may also add to the protection until its ratings are exceeded.
Getting back to the language issue, the term “Auxilliary Contactor” may be a reference to a general purpose contactor that is not intended to be used as a “Safety Device”. Or it may be talking about a secondary contactor used to disrupt a circuit should the primary device fail to open. Welded contacts are a real concern when products are not matched to the application appropriately. And sometimes, when certain products have reached the end of life, they may not fail in a desirable way. That is why there are frequently other ways to disconnect power from a device.