There is no generally accepted definition for broad band or narrow band.
In my opinion when BW/Fcenter ratio << 1, it is narrow band, from a filter or amplifier design point of view. So a 9 kHz BW at 455 kHz is narrow band in my opinion.
A 10..20 MHz amplifier (BW/Fcenter = 0.67) I classify as broad band.
When you need to design an audio filter that passes. 300...3400 kHz (BW/fc = 1.67) , I would classify as broad band, but a 390...410 Hz filter (BW/fc = 0.05) I would classify as narrow band.
You will sure find people that would classify 10 kHz at 150 MHz a narrowband channel , but 100 kHz at 150 MHz a wide band channel.
From a signal perspective one could look to Fup/Flow ratio (BW = Fup - Flow). If this is << 1, output wave forms generally have a sinusoidal envelope, so you can use steady state analysis to design the circuit. You may classify that as narrow band, as you use design rules applicable to tuned amplifiers/filters. When Fup/Flow > 1, the output waveform can be anything and then a transient approach would be preferred.