Thanks for taking the effort to make this video.
I have a few observations:
1) Are you taking into account the stray capacitance of your inductor test jig? Is the bottom layer split as well as the top, if it is double sided PCB material? Otherwise you have a PCB capacitor in parallel with your DUT. Also you introduce a small amount of stray capacitance with the PCB material extending past the banana plugs. Basically you've made a small dipole antenna.
2) Did you perform an open/short calibration of your LCR meter prior to taking measurements? On top of that, you switch from the PCB test jig to the tweezer probe without recalibrating the meter. Can you see where this could cause accuracy problems with the tweezer?
3) For me, one of the most useful purposes of the LCR meter is that it gives me the ability to pick specific devices out of a batch of components that more closely match calculated requirements. Try reading the values of a batch of ceramic capacitors and see how closely they measure to their manufacturer's specification. Being able to use more accurate components means filters, tanks and other tuned circuits more closely meet their design specification and give better performance in real life.
4) Gold plating on the tweezer probes does not introduce significant error. If you want to reuse chip components, clean them with IPA before measuring them to get rid of the flux or other contaminants.