I am little confused here, the inductance is fixed independent of frequency, but the impedance is dependent on the frequency by jwL. So, the meter is measuring the inductance or the impedance?
Ignoring the effect of the iron core, yes, inductance is independent of frequency for a perfect inductor. And the meter will measure inductance not impedance, although the LCR40 or similar will also tell you the DC resistance of the coil.
so, VL = L dI/dt
Where VL is the pd across the inductor, L is inductance in henries, dI/dt is the rate of change of current through the inductor in Amps / second
So, when a current changes through an inductor, there is a voltage produced across the inductor. The relationship between the rate of change of the current dI/dt, and the induced voltage VL is what we call the inductance L. For a perfect inductor that does not change its properties with the rate of change of current, then L is a constant. The units of inductance are henries.
Now XL = 2 π f L (or as you have said, XL = jwL )
Where XL is the reactance of the inductor in ohms. f is frequency in Hz. Reactance, as the units indicate, is the "resistance" of the inductor to the AC current flowing through it, and for instance, if you had a pure inductor, you could use ohm's law to calculate the current that would flow for a given voltage across the inductor. I = V / XL
But of course the current due to the inductance is 90 deg phase shifted to the voltage, and the current due to the DC resistance is in phase with the voltage, so that leads to all sorts of other considerations and some vector maths. That is why we have to think about reactance being different from inductance.
I have hit a wall here in not being able to use a full set of maths notation in this forum, so to cut a long story short, if you haven't wandered off for a cup of coffee already, the overall impedance of an inductor is the vector sum of XL and R. And we can use this to calculate the overall current with Ohm's Law.
Hope this has helped.