With a hand-held light meter the photographer must decide how he/she wants to measure the light in a given scene. Hand-held meters can often measure either reflected light (light bouncing off a specific portion of the scene) or incident light (light striking the general area). An in-camera meter is measuring reflected light from pre-defined areas in the scene. The camera meter setting (average, spot, matrix, etc.) determines which metering mode algorithm the camera will use and how it will 'weight' the readings from the various areas in the scene. Using a hand-held meter those decisions would be up to you to determine. You would set the exposure manually and ignore the camera determined setting - which may well be different - so how the camera metering mode is set won't matter. In a tricky lighting scenario an experienced photographer may well get a better exposure w/a hand-held light meter; however, it typical situations modern in-camera metering systems are very accurate. Using a hand-held meter is a great way to understand how your camera evaluates a given scene.
Camera's metering modes use the camera's internal meter and know nothing about your external one. Unless you know what exactly you are doing (and what for), put your camera into a fully manual exposure mode (including manual ISO) when you are using an external meter.
An internal meter and an external meter can be used together if they measure different kinds of light (such as the internal meter meters the ambient light and the external meter is used as a flash meter).