I’ve not seen that definition of inflation. I think you’re thinking of some type of equilibrium/market efficiency theory, where to support higher prices there has to be income growth, which is false especially in the short term where prices can rapidly rise faster than income could realistically keep pace and consumer is just worse off.
It's the standard definition, where inflation is a change in the price level, and the nominal GDP is defined as the real GDP times the price level Y×P. Since GDP is a just a sum of income components, of which one is wages, an increase in the price level implies a proportional increase in wages.