Definitely worth a punt on the birth certificate in my opinion!
The only Matthew Lawrence I can find on the census who is remotely like a farmer is an ag lab in Basford in 1881, aged 63, with wife Elizabeth aged 70. Following this couple back through the censuses, in 1871 they are in Southwell Notts with children Ann (24) and Ellen (17). In 1861 they also have a son William born c. 1852 in Borrowash, Derbyshire.
Maybe worth following up Ann and especially Ellen? Perhaps you are dealing with a name change here, possibly because of an illegitimate birth and later reversion to a birth name rather than a step father's.
Incidentally, there is a marriage of a Florence K Jackson to a John Moore in Basford RD Sep 1912. I've no idea whether it's relevant or not, but maybe something to consider.
Good luck!
Anne