Your "L12-100-100-12-I" per the below from the data sheet probably has the I option for internal analog position feedback. This would usually be an internal pot similar to the pot in a servo. The white wire probably is connected to the wiper on the pot and would output a voltage between the white wire and the black ground wire relative to the position of the actuator. Hopefully you haven't damaged the pot (probably not of the white wire was only connected to an arduino PWM pin)
S 2-wire open-loop interface (like B option) with limit switching at stroke endpoints.
P Simple analog position feedback signal, no on-board controller.
I Integrated controller with Industrial and RC servo interfaces (see L12 Controller Options section). Not available with 10mm stroke length configurations.
R RC Linear Servo. Not available with 10mm stroke or 12 volts.
L: Mechanical or electrical interface customizations
Custom option codes will be issued by