Depends on where you understand "widely recommended". This actually refers to operation from batteries which are not capable of adequately powering the servos where a dip in the supply will clearly cause the Arduino to fail. And yes, the same requirement would apply to the use of a mains supply.
If the said supply is capable of powering the servos and the servos actually operate at 5 V, then you certainly should power both from the same supply - but you still need to connect the power for the servos (5 V and ground) directly to the power supply output terminals and do the same for the Arduino. This then means the servo control lines must travel along with the power lines to the power supply where they then travel back to the Arduino together with its power lines.
As to the motion sensor, if it is as you suggest here, an open-collector pull-down, then it should be isolated from the Arduino by a diode, cathode to motion sensor and anode to Arduino, with the 1k pull-up resistor on the Arduino side, to the Arduino 5 V. This of course means the motion sensor can pull the Arduino input LOW, but never supply any current to it. Same for any open-collector input device.
The 1K may be more appropriate than using only INPUT_PULLUP if the sensor is at some distance with cabling in between.