Self-resonance frequency of chip inductors
I answer to the questions in the OP in order


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.
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/2\pi\$ and the parallel  resonance frequency \$f_p=\omega_p/2\pi\$: perhaps you could look into modern VHF/UHF electronics applications if this characteristics is of some use (for example in the design of filters)