John1
There is no doubt the correct name was Perseverance Inn.
With the sole exception of the 1891 Census, in every record I have viewed from the 1852 O.S.Map to the 2017 Planning Application for change of use (both of which may be considered as authoritative), the name of the Inn is Perseverance.
Further, as the enumerators of the 1881 and 1901 Censuses clearly wrote ‘Perseverance’, it is doubtful it was known locally as the Providence in 1891.
The most likely explanation for the confusion is that the enumerator simply made a mistake.
John.