Yes, I know about all the above websites I have posted them here many times. A old friend of mine Dr. Kelly is the developer of
http://fermi.jhuapl.edu/denscalc.html
You are also getting ahead of yourself

So, it will take longer to explain it all

Try and slow down.
I could have posted all those but wanted to stay simple but I'm glad to see you did some searches

. Here is the original Kelly with Sigma-t built in.......read below first, as to what this really is.
http://www.phys.ocean.dal.ca/~kelley/seawater/WaterProperties.html
Sigma-t
There are two types. Your meter is not using true Sigma-t. In seawater manuals and the one on the link and the Sigma-t's on the links you gave are DENSITY Sigma-t. Density DOES NOT = Specific Gravity. You meter is using SG Sigma-t, which is bad, as the name is confusing some people. The real Sigma-t is a term in seawater chemistry for DENSITY. So, I now need to explain these two so you do not get lost.
Density = Mass / Volume and in seawater is often expressed as Sigma-t. Sometimes the Sigma-t value may or may not be brought to is say 26, as in the above Kelly Calculator, where it reads say 1026.00
SG = weight of a volume of a sample / weight of an equal volume of pure water.
As one leaves the max density of water of about 4C and the temp increases the Density and SG begin to seperate in their values. The higher the temp the less equal they are to each other.
One thing on SG is there are often set points or calibration points. For example, a hydrometer may be calibrated to 4C, 15C, 20C, 25C etc. Hydrometers can not automatically adjust themselves to temp but EC meters can. Example; A std NSW bulb type floating hydrometer is calibrated to 15C and = 1.0264 SG = 35 ppt. If you now take that bulb type floating hydrometer and put it in 25C water it will read 1.0251 =35 ppt and it's Density @ 25C =1.0234. We call this temperature correction. We can not have this meter read 1.0264 in 25C water. If it did read that the salinity would not be 35 ppt NSW but 36.7. Most floating types in this hobby are calibrated to 25C/77F, so if it is in 25C a reading of 1.0264 = 35 ppt NSW.
I could not find one single scientific study that suggests a salinity of 35ppt is equals to a density of 1.026 at a temperature of 25C at the surface of the ocean
You won't , as the std SG is @ 15C and the hydrometer is calibrated to 15C. They do not use LFS type hydrometers calibrated to 25C. If they did use one then it would = 1.0264 @ 25C
35ppt is equals to a density of 1023.3 at 25C
Yes, Density and that is not SG which would be a 1.0251 SG.
"Also, look at this calculator:"
http://www.saltyzoo.com/SaltyCalcs/SgPptConv.php
That is for LFS hydrometers and is fine for those. 99 % of the people in this hobby use those 75-77F and not a 15 C
a salinity of 35.0 equals a sigma-t of 26 (thus SG of 1.026) but this is at 15C!!! That explains why I get the same results on this table when I set my meter to sigma-15:
It appears the meters is being set to 15C std in this mode and not doing auto corrected temp
"Practical Salinity Scale (PSS), 1978: Definition
Yes, I know all that.
Firstly, the new PSU scale is defined in terms of conductivity - NOT solutions. .
The solution is what gives the conductivity. It is right here
potassium chloride (KC1) ****solution****, in which the mass fraction of KC1 is 32.4356 x 10-3
Secondly, the equation for salinity has temperature all over it in such a way that it adjusts the value for temperatures other than the reference temperature of 15C - nothing cancels out
Yes, but your meter has a built in equation to adjust readings based on the temp of the water sample. It is called ATC (Automatic Temperature Compensation). If it did not have this and some really cheap metes do not, then all you have posted must come into play, as it than must be calibrated at 15C. If the temp is other than 15C YOU must use an equation to get the right reading for the higher sample temp.
The Great Barrier Reef were found to have salinity of about 34-35PSU over a certain period of time. Is salinity always quoted at the reference temperature of 15C then? If true, then the actual salinity of the GBR is then not 34-35 because the temperature of the GBR is not 15C - but more like 22C - 25C.
No, chemical oceanographers always ref. things back to std, which is 15C, that way all are on the same page. If you calibrated your meter @53 mS in ATC mode and put it in the same place they did, it will read the same 34-35PSU. The meter is correcting back to 15C std, correcting for the temp off-set and giving the proper reading, as it would, if the sample was at 15C.
(as Craig pointed out salinity is a measure of the solute content in water, and will not change with temperature - you're measuring the specific gravity -- a weight ratio -- as a proxy for salinity) at roughly 35-36 ppt (parts per thousand) and your temperature at the lower end around 80F and the upper end around 84F."
True actual salinity is = 35 grams / l and that can not change with temp. The meter can not measure that. So, a electronic scheme was developed , base on conductivity of a solution, called a Conductivity/EC meter, which by electronic math conversion makes it as if was grams of seawater.
Secondly, the equation for salinity has temperature all over it in such a way that it adjusts the value for temperatures other than the reference
Yes, your meter does that for you with its ATC