If you have a camera with resolution Y x X, then is active pixel count always the product of the resolution? Are they saying essentially the same thing?
The term resolution means several things depending on the context used. In digital cameras it is recommended not to use the term resolution, but "effective pixels", or "recorded pixels".
If your camera has a sensor of X x Y yes, the total pixels are the result of the product of X x Y, but they are not the same thing, because the unit "Megapixels" is not as accurate, but a round number; and it does not give you the proportions (aspect ratio) of your sensor either. So basically it gives you less information.
Sort of. Technically, no, but in common usage, often yes.
Let me explain. :)
"Resolution" means "the amount of detail which can be resolved". In computer terms, this is often directly connected to the number of pixels used in a display, and because of this, the terms became a little confused. If your screen is the same size, and pixels are all there are to it, the pixel dimensions of the screen effectively define the resolution. And in computers, we've gotten into the (objectively bad, but it's hard to fight the drift of language) habit of calling an images' pixel dimensions its "resolution".
Because you mention "active pixel count", I think you're asking if the pixel dimensions of an image from a camera correspond to the number of pixels on the sensor. And yes, they roughly do — see Why are effective pixels greater than the actual resolution? for some discussion of the non-active pixels.
But it's good to be careful about the word "resolution", still. The overall actual detail resolved — the true resolution — is a product of (and limited by) all of the different elements in the system, from the pixel count to the amount of noise to the characteristics of the lens used to how steadily you held it. So, I wouldn't say that pixel count and resolution are the same thing, without qualification.