The term "exposure" is used for a number of different but related things in photography. I can see how this might be confusing. Here are six different ways in which it is used:

1. The combination of all factors which make a photograph have a certain overall brightness. The key factors are shutter speed, lens aperture, and sensor or film sensitivity (a.k.a. ISO), in combination with the light levels in the scene itself. Sometimes, post-processing can also be involved, as in "pushing the exposure" when developing film or with an "exposure" slider in a RAW development program. This is the sense used in the common phrase "exposure triangle" (which I don't recommend, but for other reasons).
2. The amount of light allowed to to fall on an area of the sensor or film. This is related to the previous sense, but is more strict. It's arguably more technically-correct, but I think the wider definition is more common these days. Here, though, ISO sensitivity (and definitely post processing) aren't included — it's the light in the scene combined with the size of aperture you use combined with the time the shutter is open. In this sense, one might say that the same exposure yields different results at different ISOs.
3. Exposure time — A synonym for shutter speed. Or, technically, exposure time is the result of a certain shutter speed. Sometimes one might hear just "exposure" for this, but usually this sense is qualified with the word "time", or in combination with some unit of time (as in: "Exposure: 30 seconds"). "Shutter speed" is itself jargon, since we don't really mean the speed at which the shutter travels.
4. The result of a single click of the shutter. An HDR image, for example, may be said to be composed of multiple exposures — several images taken in a row.
5. The independent effects of two different sources of light within a single shot. For example, in the "dragging the shutter" technique, the resulting image may be said to combine the flash exposure and the ambient exposure — two "exposures" overlaid in the same image.
6. Occasionally, a term for a photograph, usually used in a semi-technical or jargon-ish way. This is often referring to unsorted, unedited negatives or proof prints, or a collection of images just dumped off a memory card. Or, even potential photographs, as in "a 36-exposure roll of film".

Exposure value is a common technical term which uses the first sense in the list above. You can read more about that in the question What is the EV scale?.

Confusion between sense #1 and sense #2 above sometimes results in long, pedantic flamewars between people who both think the other person is wrong, and don't realize that they're actually talking past each other.

In the example sentence you give, "If you use a quick shutter speed, you can just raise the exposure to compensate", I think the term is simply accidentally misused and that they probably meant "ISO sensitivity". Raising the exposure (or exposure value) doesn't really make any sense in that context.