So I have a bulb and when it is fully started on my lantern one filament is super bright purple and looks like it's making purple arcs. However I'm not sure if it is being run cold cathode because it's not dim.

Cold cathode means the electrons need extra kick to get emitted, the heat alone is not sufficient to let them freely escape the surface.
It could be because the cathode runs really cold (because of too low current or too short time after power on during startup), but also when the surface has too high electron liberation energy so the heat is not enough. The second usually happens when the emission material is spent, so all the electrons have to be emitted by the bare filament wire.