I think every time family historians say or write either of these two words they should take a deep breath and stop.
Assume (dictionary definition = to take for granted; to claim unduly)
Presume (dictionary definition = to take for granted; to suppose to be true without proof)
I cannot presume or assume anything without the evidence to do so. Yes you can get factual inaccuracies in any record including censuses and certificates, but at this point the 1861 census is the only evidence we have for Mary's origins (the rest as you've stated before is assumption and not evidence). Until you explore the evidence you do have, which is a possible Cumberland origin and not in itself unreasonable, it is not really possible to know whether it is a red herring?
When we assume and presume we should do it on the basis of known probability. In asking me the question whether I would presume Talkin Tarn is a red herring, in reality you are asking me to calculate the probability. I don't think I have enough sound evidence to make the calculation and therefore you are really requesting me to speculate - the third word in family history to watch out for. It is not so often said, but it is very often poorly done.
Speculate (Dictionary definition = to make theories or guesses; to consider a matter from every point of view).
I've tried here to answer your question using the second definition of the word and not the first, but in reality as I have said before I have very little actual evidence to go on. Having said that why would someone (if the enumerator has not got it wrong and he might very well have) pick 'Talkin Tarn' of all places to say they were from. If you weren't born there what would be your chances of ever having heard of the place?
You really need to follow the lead through before you can dismiss it as a red herring.
As to why Mary was easier to find in 1861 then the earlier and later censuses, it is probably dependent on her circumstances. If she was living with a man in 1851 then she and her children may very well be there, but in another name. Ditto the later census but include the possibility of her death. If anyone was in an institution at the time of the census then they will often merely be enumerated in their initials. Mary could have been missed off the 1851 census when life was perhaps much more of a struggle for her (many people paid their rent daily and moonlight flits ahead of the rent man were common - and if the rent man couldn't catch families, the census enumerated would have had little chance). The poor living at the margins were and are often missed off censuses. I would be thankful, under the circumstances in a very large city, that the census enumerator caught Mary at least once, in the surname and place you expected her to be.
If you have Mary Jane's birth certificate have you searched the address and the surrounding area on the 1851 census, concentrating on any possible candidates no matter what there surnames are?
Regards
Valda