What definition of "seamount" is used by the largest utilised databases of seamounts?
The largest database of seamounts contains over 33,000 seamounts and was created by Yesson, Chris; Clark, MR; Taylor, M; Rogers, AD.  Link

For their list, they defined seamounts and knolls as 


  Seamounts and knolls are 'undersea mountains', the former rising more than 1000 m from the sea floor
  Yesson, C et al. (2011): Lists of seamounts and knolls in different formats. doi:10.1594/PANGAEA.757564


In their paper, they set the definition of a seamount as a sub-surface feature, not breaking the water.  This puts the highest elevation in their dataset is 2m below sea level.  The pulled their definition from Morato, et al (2008), which states: "seamounts are defined as any topographically distinct seafloor feature that is at least 200 m higher than the surrounding seafloor, but which does not break the sea surface."  Then, they subdivided the features into knolls and seamounts at an heigh of 1km (with knolls being between 200m and 1km in height).

So, specifically answering the question, that databases uses the Morato, et al (2008) definition, which excludes super-surface seamounts.