Anybody have a source that defines it?
The short answer is no.
There is no universally accepted definition of a mountain. Elevation, volume, relief, steepness, spacing and continuity have been used as criteria for defining a mountain. In the Oxford English Dictionary a mountain is defined as "a natural elevation of the earth surface rising more or less abruptly from the surrounding level and attaining an altitude which, relatively to the adjacent elevation, is impressive or notable."
Whether a landform is called a mountain may depend on local usage. Mount Scott outside Lawton, Oklahoma is only 251 m (823 ft) from its base to its highest point. Whittow's Dictionary of Physical Geography states "Some authorities regard eminences above 600 metres (2,000 ft) as mountains, those below being referred to as hills."
At one time the U.S. Board on Geographic Names defined a mountain as being 1,000 feet (300 m) or taller, but has abandoned the definition since the 1970s. Any similar landform lower than this height was considered a hill. However, today, the United States Geological Survey (USGS) concludes that these terms do not have technical definitions in the US.
You may however find this bit helpful:
The UN Environmental Programme's definition of "mountainous environment" includes any of the following:
Elevation of at least 2,500 m (8,200 ft);
Elevation of at least 1,500 m (4,900 ft), with a slope greater than 2 degrees;
Elevation of at least 1,000 m (3,300 ft), with a slope greater than 5 degrees;
Elevation of at least 300 m (980 ft), with a 300 m (980 ft) elevation range within 7 km (4.3 mi).
Using these definitions, mountains cover 33% of Eurasia, 19% of South America, 24% of North America, and 14% of Africa. As a whole, 24% of the Earth's land mass is mountainous.
For me, hard rock is integral to what makes a mountain. A pile of clay, dirt, and/or sand, no matter how high is not a mountain.