First of all, your date is wrong.I have seen the 18 Dec. 1755 date before, and suspect that it arose from two different errors.For the year, 1755 is the date found on the Greeno tombstone in Rutland.However, this stone was not put up until long after Daniel's death (presumably by his grandsons Benjamin and Amasa, for whom the stone appears to have been erected - Daniel's original stone, if it still survived, was probably discarded when the family plot was plowed under and the bodies removed to the city cemetery).It appears to have been derived from taking the death date, 1810, and subtracting the age at death.The problem is that they had the age right, but the death date wrong.Both contemporary church records and a newspaper death notice report his death in 1807, aged 55.Thus his birth was not in 1755, but instead in 1751/2. (As to Dec. 18, I suspect that since it is the birthdate also attributed to his son Moses, somewhere along the line someone accidentally transfered it from one to the other.)
Regarding Daniel's maternal parentage, there is no question.While his father married Elizabeth, she was never a candidate to be Daniel's mother (do you recall where you saw this?).You have notice the Emerson name connection, which provided additional circumstancial support for the identification of Daniel of Rutland with the son of Daniel from NH, but there is other, water-tight evidence.The Daniel of NH and NS (I hesitate to call him Sr., as his father was also named Daniel) was not previously known to have had a son Daniel, or any children by Hannah.However, that she had a son is proven by the will of David Emerson, Hannah's father.He leaves a legacy to "grandson" Daniel Greeno, son of "my daughter" Hannah.
Thus Daniel or Rutland was born to Daniel and Hannah (Emerson) Greeno, presumably sometime in 1752.