An electronic barometer measures the ambient air pressure. Like most every electronic measuring device, it needs to be calibrated to a known value first (Eg. you tell it "right now the air pressure is 951 mbar" , it will remember that). From the air pressure the altitude can be deducted, but only as long as the weather doesn't change. This is the second calibration involved. Since weather changes frequently, the device can optionally use the altitude calculated from GPS to automatically recalibrate the altimeter, which will keep it accurate to a few meters or so.
The Garmin 62s has 4 holes in a diamond shape under the battery cover. If you cover these with fingers, the pressure reading increases.
You can calibrate the sensor and make it read what the local weather station was reporting and then proceed to adjust it for elevation.
Barometric Pressure is the air pressure adjusted for elevation (sea level reference). Ambient or Atmospheric pressure is the raw pressure reading.
The pressure that you will get from your local weather station is also the pressure at sea level which it will work out using the elevation of the pressure sensor on the weather station.
On the 62s there is an option to do that. From the Elevation Plot screen, press Menu, then down to calibrate altimeter, enter, say no to knowing the correct elevation, yes to knowing the correct pressure, and then enter the mb from the local weather station. From the main screen, setup, altimeter and you'll have the following - auto cal=on, bar mode=fixed elev, press trend=save when pwr on, plot type=baro press