I understand the two lathe axis.
X and Z
If I want to use the spindle to drill 4 equally spaced holes with live tooling.

Is it still called the spindle or is there an axis definition that the lathe knows that I’m using the spindle for positioning?

Do you have a servo spindle? You mean holes on a pitch circle?  I think there might be a C axis but I'm not sure if Mach supports it.

On a lathe, there are typically different axes to describe different motion directions. Usually, the X-axis represents lateral movement, and the Z-axis represents longitudinal movement. The main spindle is typically used for rotating cutting tools or drills. In the scenario you described, if you are using the main spindle to drill 4 equidistant holes, the main spindle is still referred to as the main spindle, but specific axis definitions or work coordinate systems may need to be set in the program so that the lathe knows you are using the main spindle for positioning.

By setting axis definitions and work coordinate systems correctly, you can ensure that the lathe understands how to interpret your operational commands, allowing the main spindle to accurately position and perform the required operations. This helps ensure the accuracy and consistency of workpiece machining. If you have any questions about the operation and programming of a specific lathe, it is advisable to consult the lathe's operation manual or seek guidance from relevant professionals for more detailed instructions.