I approach these sort of problems by having a "theory of the case". It may be incorrect, and as more information comes in I can change or abandon the current theory for another one.
In this particular case, my own present best guess (theory) is that Matilda was illegitimate, and the varying father's names and occupations reflects her way of dealing with that. When asked who your father was, or what was his occupation, are you going to say I don't know? In that situation, many just gave names. Again, this guess of mine may be completely wrong.
So, I would say OP should turn to DNA testing, and look for matches.
Yes Wexflyer that is good advice about about having a 'theory of the case'. Several genealogy courses I have been on have suggested this then reflecting on and noting how the theory could be proved/ disproved.
I must admit having seen the marriage certificate with the father James as policeman certainly made me think. What better way really to forestall discussion about parentage than by claiming a policeman as a parent? Also the lack of perhaps such a person day to day can be explained by the requirement that policemen did not usually live in the communities they served in.
Of course Matilda may be repeating what her mother had told her about her father and this may be correct.
I find this family fascinating, if elusive, and wonder there is more we can do with registers, logic and leaps of logic before OP goes the DNA way.
Perhaps the OP could generate some theories of the case and we could have a last crack at them.....? Perhaps summarising the new situation.....?
Also making me think is the Aghadowey saw on a facebook post.
So two theories:
1 James the policeman as father
2 the parents as set out in the FB post Aghadowey saw "Posted on a Facebook forum also where someone has replied "Matilda is interred in Derry City Cemetery, it states she was born Burt, Co. Donegal."
In the cemetery records parents listed as Robert and Jane Stephenson. She died 15 Feb.1940 and age is given as 55 (c1885)."
ETA 3 Matilda Black/Cadden looking at Matilda Black nee Daly