Don't doubt the lamp. It makes my cotton grow like crazy. Don't know about the range, but it covers at least a standard 7x7 (1 water in the middle, and 3 in each direction, including diagonals).
It may cover more than 7x7. I also haven't tested the vertical limits. I have mine about 3 blocks about the crops.
It also eats the Herbia essentia like crazy. My essence Furnace is chewing through saplings and charcoal. Not as fast as the cotton is being produced, but pretty darn fast. Maybe 1 sapling per 4 cotton.
If you don't supply it with any Herbia essentia, I don't think it confers any benefit. My cotton squares with the lamp - but powered off - and without the lamp entirely seem to grow cotton at the same rate. Pic of my setup below. The lamps are currently off. I had already accumulated over 300k cotton.