I've always viewed Integrated amps as essentially the predecessors to receivers, because we used to use external tuners. I think the inclusion of a pre-amplification stage as a qualifier is debatable as a pre-amplifier typically came just before the power amplifier and did not necessarily include more than one input channel, but in the late 70's we started to see many more pre-amplifiers that served as our connectivity as an integrated amplifier would, sans the main amplification stage.
My definition of an Integrated amp is a power amplifier that allows for switchable connectivity to more than one input.