You seem to have problems with tracing folk in Durham. Its always difficult to get to grips with a new area of research as you have to start from scratch learning the how and where to search.
the following is long (get a flask and some sandwiches to see you through) but may help to see how I went about looking
You have the indexed burial entry - there's not likely to be any further info on the image to confirm/deny its your man. so you can't confirm from that its the man you are looking for
the death registration - which will give a place of death and also the name of the informant - which could well be a known relative - may not be - but its your best shot at confirmation
FreeBMD
Search Deaths
enter the surname and first name
Age/DoB put in a range for his year of birth I used 1803-1809
use a date range of 1861 (when he was on the census) and 1871 (when the census recorded Hannah as his widow)
Specify the County as Durham
3 results within Durham for deaths in that name
Check the first two on the GRO site (which unlike the old printed index does give an age at death)
1862 Teesdale - GRO site says he's 2 so you can discount him
1863 Sunderland - GRO site says he's 64
I checked the Sunderland Registrars site for this death and they have a local register reference of ES17/123
ES is the East Sunderland sub district of Sunderland Cicil Registration District
see:
https://www.ukbmd.org.uk/reg/districts/sunderland.htmlCensus 1861 Piece3777/Folio120/Page19
has a William Hunt, a greengrocer and his wife Barbara both from Norfolk in Reg District Sunderland sub district 3 East Sunderland that says he was 63 in 1861 but . .
this is his burial register entry
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSJW-W1LW?which matches the occupation and address on that 1861 census -
So he can be discounted too
Which leaves you with that 1867 registration in Chester-le-Street and IF he died in Co Durham that is highly possible . Only that £3 death register image from the GRO would 'perhaps' confirm that.
Boo