This is exactly my understanding problem.
Pushing = more light
Pulling = less light
but you seem to be saying the opposite Stephen
You push a film when there isn't enough light to use the aperture and shutter speed you want (e.g. black cat in unlit coal cellar when you use 1/1000th second to stop the motion, and f/22 to get maximum depth of field to cover the area it's moving though). You're attempting to get a denser negative with less light than the film manufacturers recommend. If you prefer an inaccurate but more visual statement, you're pulling more light through the lens than actually exists (and breaking the laws of physics, but, hey, laws are meant to be broken, aren't they?)
You pull a film when there is too much light to use the aperture and shutter speed you want (e.g. you want a 10 second exposure time to smooth the waves on a beach at mid day in summer with a cloudless sky and use f/1.2 to minimise the depth of field to enhance the wavefront). You're attempting to reduce the negative density from a compete black. And using the same inaccurate statement, you're pulling light away because you haven't got a neutral density filter....
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Now skip the rest to avoid confusion
Or, you do the same thing (increase or decrease the development time with a corresponding change in exposure) to increase or decrease the contrast of the negative for printing purposes. That falls under the Zone System.
At the end of the day, forget the terms. You increase the development and hope to get something out of an underexposed negative, or you cut development in the hope of salvaging gross overexposure (which arguably you'd achieve better with a reducer (look up Farmer's reducer, but this opens a whole new area). Underexposure, if deliberate, is achieved by using a higher film speed than the film actually has.
And if you want to get really complex, just remember that film speeds are determined scientifically where every variable is controlled by the ISO standard to give a negative where the image starts to appear at a given density above base + fog (definition: film bases are not 100% transmissive; they cut some light; all developers produce some development even where no light has fallen - that's chemical fogging). Change the developer, change the temperature (even if you use the same 68 degrees, does your thermometer exactly agree with the standard one?), change the agitation, change the tank size/type which would affect the flow, change the time (did you stop development at exactly the same time as the standard, use a stop bath or not?), did you pre soak which would affect developer take up initially? These are reasons why you may wish to deviate from the manufacturer's rating, and that's all before we get into the need of the enlarger types for the optimum contrast in the negative (determined by the development (and developer!)).