Square Footage under Beam is the measurement in feet of an area directly under some type of covering (roof, canopy, cliff overhang). Generally referring to support beams of a structure and all of the area which it covers. In real estate it would include all areas under roof.
Square Foot Living Space is the measurement in feet of the above grade interior area of a structure that is livable with proper access, heating and in most cases cooling. In reas estate that would exclude carports, garages, attached laundry rooms, porches, patios, basements, and attics that do not have proper ceiling height. In the case of multi-level homes (2 story, 3 story) you would also exclude any open space over the lower levels.
The tax assessor goes by the builders measurements when the plans are submitted and building permits are issued. During the next assessment date they usually use the appraised measurement if there is any large differences from time of construction and resale.
When choosing comparables the GLA (gross livable area) square footage which is also the assessors square footage is used. To calculate the square footage of your own home (assuming livable) you would measure all sides of the foundation excluding any non-livable areas mentioned above and multiply length times width. In the case of a Condo or Townhouse (attached with common walls to others) you would measure the interior air space.