There are a few questions in there, so I will try to address them all. The Bueler Time Delay relays are unlike the vast majority of other relays you will find. The difference is that these relays are so-called "smart" relays. They have embedded circuitry that handles the triggering as well as the timing functions of the timed relay.
For most relays (i.e. "dumb relays"), the internal construction consists of a solenoid coil that actuates an armature which connects and disconnects a circuit (based on the type of relay). There is a standardized pinout configuration for automotive relays, and pins 85 and 86 are the "control pins". When you pass a DC current from pin 86 to pin 85, a magnetic field is created within the relay coil causing it to switch. See this video for a more in-depth explanation of how this works.
The reason why you were probably told to use a Flyback Diode on pins 85 and 86 is because that is the proper way of protecting the relay coil circuit of a run of the mill relay. However, these timed relays are designed with a circuit that handles all of this for you. It has a logic component that is capable of timing events, and it is constantly monitoring the "trigger" pin (pin 86). Using a Flyback Diode on this type of relay is unnecessary.
I believe the reason why the relay works as expected while manually unhooking the wire is because these relays trigger on a "pulse". From the information I was able to gather on these relays (see this video), the relay actuates for 2 minutes when the trigger pin is tied to 12V, not when it is disconnected. It's possible that you are causing a pulse on the trigger pin with the wire that you do not notice because it is happening on a millisecond scale.
You can test this by manually triggering the relay with a piece of wire. Does the relay switch immediately as soon as you touch 12V to pin 86, or does it wait until you remove it? If the relay switches immediately, the easiest way to trigger this relay would be to install a momentary switch (or hijacking an existing one) somewhere in the car. Just press the switch before you exit the car and the fan will continue to run for the predetermined time.
If this is undesirable, you can look into creating a single-shot circuit that fires each time you turn off the vehicle.