The matter is confused by the fact there were two Santa Claus poems in the 1820s.
The probable Moore Old Santeclaus (1821) and A Visit from St Nicholas (1823).
The latter was more popular than the mean-spirited Old Santeclaus and with Livingstone dead c.1830, the authorship was assigned by rumour to the more prominent Moore.
Moore chose not to dispute this and later included VFSN in a book of Christmas poetry without actually claimng he wrote it.