Are you aware that the word "piezo" is a generic term, and there are numerous types of devices that could be called a "piezo"? It is easier to help you if you describe your "piezo" with more detail.
JimB already asked how you made your "resonance" measurements, and I repeat the request.
Piezoelectric elements generally are designed to be resonant at some particular frequency, and their equivalent capacitance varies a lot in the vicinity of the resonance frequencies.
Your Tenma meter only measures capacitance at either 1 kHz or 100 Hz, depending on the range you have selected. The supplier of your "piezo" elements has measured the capacitance at some particular frequency; you should find out what frequency they used.
If you have determined the apparent capacitance with a "resonance" measurement and then calculated the apparent inductance of the leads using the capacitance given to you for the device, you are overlooking the fact that the "piezo" has its own resonance that has almost nothing to do with the leads.
I have measured the apparent capacitance of a piezoelectric bender of the type used to make the beep sound in portable electronic devices and attached a couple of plots.
The first plot shows the apparent capacitance over a frequency range of 1 kHz to 100 kHz. You can see two major resonances, and a number of minor resonances. Notice that the apparent capacitance varies from 1 nF at the bottom of the plot, to 200 nF at the top. Actually, the variation is greater than that.
The second plot zooms in on a narrower frequency range of 3.5 kHz to 4 kHz. The apparent capacitance varies from about 43 nF at the left edge of the plot (3.5 kHz) to 2 µF at the peak (approx 3.67 kHz).
You say "For example the datasheet says Part 1: Capacitance 683nF. Part 2: Capacitance 692nF. However I measure the values with my fairly cheap LCR meter much lower typically 50-100nF lower."
Your Tenma measures the capacitance at a relatively low frequency; it is to be expected that the apparent capacitance rises with increasing frequency as you approach resonance. You should find out at what frequency those capacitances were measured.