When was the name "cumorah" given to the hill cumorah in modern times?

Joseph Smith first visited the hill in 1823. 1827 is when he reports to have taken the Gold Plates from the hill. 1830 is when the Book of Mormon was first published. It would have been in that 7 year span.

Since 1829, the Latter Day Saints have called the hill "Cumorah", and local non-Mormons have called it "Mormon Hill" or "Gold Bible Hill". The hill has also been called "Inspiration Point".