An induced voltage (high voltage??) test proves that the transfo does not have any insulation breakdown at several times rated voltage (usually applied to the low voltage winding, between its terminals). Because any normal transfo will saturate at 120% rated, or so, voltage test has to be done at a proportionally increased frequency e.g. 150 Hz at 3 x rated volts for a 50 Hz transfo, using a special test generator.
You can do an isolation test between windings at rated frequency or DC, this would usually be described as an insulation test
"Tan Delta" usually refers to the loss angle of a capacitor. In the case of a transfo, the loss angle of the insulation between one winding and the others (which are earthed, also the core) is measured. A high loss indicates poor condition of solid insulation or oil.