A fluorescent lamp is a mercury-vapor gas discharge lamp with a phosphor coating. Doesn't that qualify as an arc discharge?

No.

An arc discharge is a hot, high current density process. [1] The discharge in a fluorescent lamp is a glow discharge: low current density, moderate temperature and extended volume. [2]

[1] https://en.wikipedia.org/wiki/Electric_arc [2] https://en.wikipedia.org/wiki/Glow_discharge

The first article gives fluorescent lights as an example of modern usage of arc lighting... The second article distinguishes between glow discharge and arcing.

And you literally need a thousand volts to kickstart a fluorescent lamp, at least on decent tubes.

Let me pull up my "flyback transformer on a chip".