NE555 Based Speed Controller circuit - Contradicting Pins in 2 Tutorials

They are both acceptable solutions.

Notice in a 555 timer, the discharge pin is simply an open collector version of the output signal.

In circuit diagram 1 the 555 timer is configured as a square wave generator. The charge/discharge path for the timing capacitor is from the 555s output which will be near top rail or bottom rail. Since, with the pot at 50%, the resistance to the cap is the same in both states, it charges and discharges at the same rate. Hence the "square wave" configuration designation.

The discharge pin is therefore surplus in this configuration, though it still switches as normal. Instead, it is used to pull down the MOSFET gate during the discharge cycle.

In circuit 2, the more traditional wiring, the charge rate is defined by R2 + whatever the pot setting is, while the discharge is purely through the pot. With this design, mid-range on the pot is not 50% mark to space. Further, during the discharge cycle, this circuit wastes considerable current ( 12mA ) through R2 for no purpose.

As such, circuit 1 is arguably the better one.

Notice though in these two particular examples, the design of the MOSFET gate driver gobbles up 12mA instead, so these two circuits are pretty close efficiency wise. A better gate driver circuit would fix that.

Personally, I would have done it this way. I might even add a nice LED in parallel with the motor so I could tell how hard the controller is driving.

I'd need to check if both end points of the pot worked as expected.

On the 555, pins 3 and 7 are very nearly equivalent to each other, with the only difference being that pin 7 is open-collector, while pin 3 has a totem-pole output. In both of your circuits, a 1K resistor is connected to pin 7 to provide the pullup function, effectively eliminating that difference.