Question about the switching behavior of a transistor as a switch
A run through SPICE shows that the delay is real. Delay depends upon the transistor characteristics. The transistor does turn off well after the base voltage drops quickly to zero. In your circuit, the collector current would be of interest, since LED brightness is directly proportional to current flowing through. A plot of collector current does show that collector current falls toward zero about 0.6 uS after Vin drops to zero, in the SPICE simulation:


Notice the small step in base voltage at the 1.0 uS mark, when the transistor is told to turn off (just like your measurement). At this same moment, collector current pulses up above the 12 mA mark, because of base-to-collector capacitance. This short pulse is difficult to see in the blue plot of Ic.
MOSfet switches don't have this storage delay, but their very large capacitances must be charged and discharged - requiring large pulse currents.