I think there is some misunderstanding about my thoughts on the use of water as the other fluid. I do not think the tubes should contain water. They should contain air (or hydrogen of helium). I updated the image in #13 to show the rest of the math. The bottom line shows how one would calculate the apparent specific gravity (what brewers use) from the differential pressure reading. Note that it depends directly on the differential pressure reading divided by the distance between wort levels in the tubes conveying pressure to the pressure sensor (carboy arrangement or conical). Thus if the pressure instrument is in error by 0.1% the SG reading will be in error by 0.1%. The nominal pressure difference depends, of course, on the delta L - the separation between the liquid levels. If that is about 27" the pressure difference will be about one psi (for water) and thus your instrument needs to be able to measure to an accuracy of 0.001 psi if you want accuracy of 0.1% in SG which, as SG is about 1 is 0.001 SG. For 13.5" separation you would have to measure to 0.0005 psi.
If you use a water filled manometer with 27" separation then, unsurprisingly enough, the height of the water column in the manometer will be about 27" and you have to be able to read that to 0.027" to achieve the 0.1% accuracy goal.
Now the other factor in apparent SG is 1 plus the ratios of the density of the gas in the tubes divided by the product of g, the density of water at the reference temperature and the separation. What I am suggesting is that this factor be set to the pressure difference when the setup is filled with water as water has apparent SG = 1 (to 3 decimal places). There is no need to measure or obtain a value for g and no need to know the density of what is in the tubes or indeed the density of water at a particular temperature.
In the conical case there is one potential problem with doing things this way if the tank is spunded and appreciable pressure applied. If, for example, 1 atmosphere CO2 is allowed to build in the head space the air in the tubes will be compressed to approximately twice its density. At atmospheric pressure the 1 + density ratio terms is approximately 1.001. With an additional atmosphere's pressure in the headspace that would increase to 1.002. That's really only 0.1% change so perhaps not a problem after all but the idea of using hydrogen or helium as the gas in the tubes was prompted by consideration of this.