I am actually quite interested in the balance between accurate meaning "correct and verifiable" and accurate meaning "precise".
A recent example occurred when I was entering the death date for one of my relatives. I have found the registration of the death (in the UK) which shows it was registered in the October to December quarter of 1972. I also have access to the Ancestry tree of the daughter of this person, which shows the date of death as 23 December 1972, but I have not found any independent confirmation of the precise date.
I debated for a while whether I should enter the precise date quoting both the death index and the Ancestry tree as sources. However given the strong opposition I have seen to using Ancestry trees as sources, which I can certainly understand and sympathise with, I ended up entering the death date as "about November 1972" and just quoting the death index, which is the maximum accuracy that any other researcher can verify (assuming they can't find a source that I missed, in which case they can update it). This is more "verifiable" but less "precise", so whether that makes it more "accurate" or not I'm not sure.
I'm sure this won't be the last time I am faced with a situation like this, so I wonder what approach other people normally take?