I don't think this has anything at all to do with Finnieston in Glasgow where there is the big crane outside the Scottish Exhibition Centre.
You say the 1841 census at Westquarter lists Davidson Sinclair, 35, born in the county. In other words, probably not in the town or parish of Lanark, but somewhere in Lanarkshire. Also ages in 1841 were supposed to be rounded down to the nearest 5 years, so his actual age could have been anything from 35 to 39. Therefore he could have been born any time from 1801 until 1806.
Then you say the 1851 census lists Davidson Sinclair, age 47, and sister Martha in Westquarter in the parish of Glassford, and saying they were born in Glassford. So he was born 1803/4, and in the parish where he was living. This is totally consistent with the 1841 census.
Then in 1861, basically the same information, only saying Finnestown in Lanarkshire instead of Glassford as the place of birth.
As all the three censuses are consistent about his age and his county of birth, it seems unlikely that either of these pieces of information are incorrect.
Therefore the inference I would draw is that Finni/esto(w)n is, or more likely was, a farm or house in the parish of Glassford. However there's no place listed in the 1851 census of Glassford with a name remotely resembling Finnestown, so it must have changed its name, or disappeared, before 1851.
By the way Westquarter is the village now named on modern maps as Glassford.