Light meter and exposure meter are interchangeable terms for the same instrument. We can use them two ways. A reflection reading is taken by pointing the meter at the subject. We are measuring the amount of light energy that is reflected from the subject -- it has already hit and is being reflected.

We can also measure the light before it hits the subject. We point the meter back at the camera from the subject's position. We are measuring the light that is about to hit. In the jargon of photography, this is called an incident light meter reading. The word "incident" is Old French for about to happen. The incident light meter is equipped with a translucent sphere or a concave sphere covering the entry opening. A incident meter reading is essentially the same as a refection reading provided the reflection meter is measuring at the subject plane and reading the reflected light from a gray card with an 18% reflective surface.

The light meter / exposure meter used in photographic applications outputs data unique to setting the exposure of a camera. We are talking aperture settings (f-number) and shutter speed intertwined with the sensitivity of film or digital chip (ISO) and of course the illuminate level.

There are non-photographic applications for the light meter (not exposure meter). These are used by architect and lighting technicians. They are dealing with lighting levels at the workplace etc. They use light meters that output data in terms of Standard Candles per square foot or square meter or LUX or Lumens.

It is possible to convert the data outputted by either style. Conversion charts and conversion factors exist so it is challenging but not impossible to make the conversion.