I noticed sometimes aircraft taxi with the logo light illuminating their tail, while others have it off. So is turning on the tail light at the discretion of the pilot?
There can be several lights on the empennage (tail section) of an airplane. There are two as shown here:
The red light is called aircraft beacon. The white light is called strobe.
When to turn them on is explained here:
All aircraft built after 11 March 1996 must have an anti-collision light system (strobe lights or rotating beacon) turned on for all flight activities in poor visibility. The anti-collision system is recommended in good visibility, where only strobes and beacon are required.
Another one is logo light:
This light illuminates the livery design on the vertical stabilizer.
For more details, please see this question:
When to use Beacon, Anti Collision, Strobe, Logo and Navigation lights?