Driving a single-relay board with Raspberry Pi
Looking at the photographs of the relay board, it looks to me like the transistor is acting as a high-side switch for both the relay and the LED, since one of the transistor leads goes directly to "+" and one of the relay coil pins goes directly to "-" (the ground plane). Therefore, the transistor would either be a PNP BJT or a P-channel FET. I'm betting on the PNP.

This would mean that you need to ground the "S" pin to activate the relay, and you need to open-circuit the pin, allowing it to float all the way to 5V, to deactivate it. Simply driving it to 3.3V will still keep the transistor activated. Although, I admit, this doesn't explain the changes in the LED, unless the voltage across the relay coil is changing enough to cause the LED to dim without allowing the relay to drop out.

In other words, the relay board is designed to behave much the way a bare relay coil would work (with low-side switching), but without requiring whatever's driving it to sink the full current of the relay. This makes a certain amount of sense.