Is there a standard definition of the term "mountain"?
There does not appear to be an agreed international definition of a mountain.

In the UK there is a colloquial definition that a mountain must be "a thousand feet high". This definition is sufficiently firmly established in a "folklore" sense that the 1995 film "The Englishman Who Went Up a Hill But Came Down a Mountain" centred around it for its plot, and this plot is allegedly based on an older story in the region.

According to the UK's Ordnance Survey,


  There is no precise international definition of a mountain... However,
  it is generally agreed that the minimum height is 610 metres (2 000
  feet). (Source)


This definition does not make clear whether a 2000' rise from the surroundings is required, or whether a peak must simply be 2000' above sea level.