Self-resonance frequency of chip inductors

I answer to the questions in the OP in order

1. Yes, the inductor frequency resonance is the first, lower value one. Its importance is due to the fact that it represent a sort of figure of merit of the component: it is the largest frequency below which the impedance of the device is inductive, i.e. increases with increasing frequency.
2. Perhaps no, since I am not aware of any use of this higher resonance frequency. However, for quartz crystal resonators two nearly equal frequencies are usually specified: the series resonance frequency $f_s=omega_s/2pi$ and the parallel resonance frequency $f_p=omega_p/2pi$: perhaps you could look into modern VHF/UHF electronics applications if this characteristics is of some use (for example in the design of filters)