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Full disclosure, I'm not a scientist just a person on the internet, but here is my understanding.
The confusion starts with our use of units which are otherwise static and predictable in "close" cosmological terms, in this case:
How do we measure a distant object and determine its distance? By measuring the light that is emitted by that object and seeing how much it has red-shifted (with the wavelength of that light being the underlying thing being measured) with an Earthbound observer as the relative point of measurement. A longer wavelength (into the red end of the spectrum) denotes the object traveling farther away from us. This last point is right in light with special relativity.
However, what if there is another thing beyond special relativity's effects also increasing the wavelength of the measured light. Our measurement becomes polluted by this other variable. That other variable is the expansion of the universe further lengthening the wavelength of light. Essentially the distance the light is traveling is being extended causing the additional special relativity effects to our sample. So since our measurement of distance is based upon the behavior of light traveling over a distance, and we derive that distance from the parameters of the measured light, we can (and must) subtract out the speed of light from a measurement and the difference we see allows us to measure the expansion of the universe by itself.
Its a spacetime effect. Take a partially inflated balloon. Draw a circle on it. Draw a line across the diameter of the circle. The diameter line represents the speed of light from one edge of the circle to the other. Starting from one side of the circle, measure 90% of the diameter across and draw a dot. Go from the other side of the diameter and again measure 90% across from to the other side of the circle across the diameter. Now blow the balloon up to twice its previous partial size. Both dots are still at their relative 90% of the diameter. So the length of the diameter line you started with is actually the combined value of both the speed of line as well as the expansion of the balloon. The place where this example falls apart is that if you were to instantly transport (impossible in our understanding of the laws of the universe) to that balloon light year section of space, you would not see the effects of the expansion of the universe. It would look like the original partially inflated balloon you stated with.