Lathe axis are confusing.

As magicniner says, inline with the spindle is the Z axis.
Standard across the spindle line is the X.
Y axis is typically if the entire X/Z axis can be moved (typically it won't be at 90 deg to the X)
C is typically if the spindle can be positional controlled.

Off course, by the time you get into that level of complexity, you're verging on mill turn, in which case it really does vary.
I just tried searching for a suitable diagram, and have just learnt nobody really uses an A-axis on lathe (should be rotary around the X axis), and B-axis which should be rotary around the Y axis, is more likely to be used for the tailstock/second spindle housing linear Z movement.

However I'm sure I've seen diagrams where the tool holders are mounted on rotational axis labelled A and B.