Simplest, cheapest way to detect if a motor is running?
Since you don't care about motor current the simplest method would be to monitor an auxiliary contact of the motor control relay. This could be used to feed to the logger.

Arduino devices are not industrially hardened and will be susceptible to EMF, etc., won't panel mount easily, etc. You also have the problem of real-time clock / calendar and extracting the data from it. Better options would be:


Industrial data logger.
Mini-brick PLC. This is the most flexible as they have mains or 12 - 24 V inputs, network capability, SD card for data-logging, etc. Manufacturers include Siemens, Allen-Bradley, Crouzet, Mitsubishi, ...




Figure 1. The Siemens LOGO! range has mini-HMI, keyboard, Ethernet, 8 inputs, 4 outputs, micro-SD, etc.



Figure 2. The Logosoft Comfort programming environment includes a graphical editor and simulator. Source: PLC Compare.

Cost of the hardware is about €120, I think. Check if there is a free download of the software.

The advantages of this approach are:


Industrial hardware - robust, finger-proof terminals, mains isolation, etc.
Can be properly documented and supported by other engineers (who won't thank you for Arduino code).
Integrated clock-calendar, network, HMI, logger and keypad.




If you insist on going the Arduino route then have a look at Industruino. They've industrialised the device to the same form-factor as the LOGO! above.



Figure 3. The Industruino packages an Arduino, display and I/O into a DIN-rail mountable package.