Another term that may sometimes be synonymous is 'cogeneration' in which the power plant produces multiple forms of energy (e.g., electric power and steam), and where both are raw-materials for a related industrial process. Probably the most classic example is that of a paper mill. Boilers produce steam. The steam passes through a turbine that spins a generator to produce electricity. Exhaust steam from the turbine is then used as a source of heat to dry freshly-made paper before is is finally condensed into water and returned to the boiler. The boiler itself burns the bark that itself cannot be used to make paper and would otherwise be a waste material. In addition, the process of making pulp produces a chemical waste called "black liquor' that can also be burned as a fuel in a boiler.
Captive power plants don't necessarily have to be islands that are disconnected from 'the grid'. In fact, it is often the case that the demand of the industrial process exceeds the capacity of the captive plant, and power must be taken from the grid to make up the difference. Also, there must be some provision to 'bootstrap' the integrated process into operation - often this means relying on grid power to start-up the plant following an outage. And it is possible that there are times when the captive plant will produce more power than can be consumed in the industrial process, and rather than throttle back the excess is sold to the grid. �