I guess what I'm saying is that if you want to compare the power that an actual example of these lamps is consuming with what it says on the tin then simply taking a current measurement isn't going to give you a meaningful answer.
If the supply was DC then it would be fine to multiply the volts by the current & call the answer Power. That's how its defined for DC.
AC isn't quite so simple.
For a simple resistive load (say a heater or an old fashioned filament lamp) then the current it takes from the supply is in phase with the voltage (ie. If you looked at the voltage and current waveforms on an oscilloscope then you would find that they tracked each other). For an inductive load this is not true. The voltage and the current will not be in phase anymore. This means that the simple act of multiplying voltage and current will not work anymore.
There are three kinds of AC power: Reactive Power, Apparent Power & Real Power.
The only one you can measure with simple meters is apparent power. Apparent power is measured in VA and is the simple product of volts * amperes.
Real Power is the 'work done' by a system (heat generation, light generation etc) and is measured in Watts. You cant measure this directly.
Reactive Power is taken by inductive & capacitive loads eg. motors, transformers etc. Reactive Power is measured in VAr. A motor will draw both reactive & Real power - the reactive part creates the magnetic field, the real part creates the torque on the output shaft & the heat within the motor. If you where to measure the volts across & the current into the motor you would be measuring the apparent power.
These three are all linked by the power triangle, whereby real power is the horizontal base, reactive power is the vertical & apparent power is the hypotenuse. The power factor is the cosine of the angle between the apparent & real powers. It is also the cosine of the angle between the voltage & current waveforms.
The relationship can also be expressed as pf = real power/apparent power.
In order to correctly measure the real power (Watts) you need a wattmeter. This does more than simply multiply the volts by the amps, it also takes account of the phase difference between the two.
I hope that's not got too heavy on the AC theory
The quantity which you can measure and compare easily is the VA. That will be of some value but it isn't the same as the Watts. This is why you are struggling to get the same value as the manufacturer. Inside the LED lamp will be a small switch mode power supply unit. The current waveform which this takes from the mains will not be sinusoidal. It will have a reactive content (due to the capacitors) and will probably be confusing your meter. The manufacturer will have used a calibrated Wattmeter to make his claims and this will measure the voltage & current as well as the angle between them.
The closest you can get, without spending silly money, would be to obtain one of the better plug-in type power meters. You need one which measures the voltage as well as the current (the cheap ones just measure current & assume volts - these will be out for the same reason that your tests are). These will also show VA & VAr as well as Watts & power factor.