NavList:

Jacques, you wrote:
"1) the formulas used for parallax correction was the same ;"

If you mean the linear term, or the "principal" term, which I think you probably do mean, then yes, I agree with that, and that's what marks these as relatively small variations on a theme.

And:
"2) Lyons proposed in NA 1767 tables for refraction correction, but not very practical ; so, Norie had proposed "linear tables", but added another correction given by table XXXV (quadratic correction)."

Well, I think you've skipped a step there. Lyons is also generally credited (e.g. by Maskelyne) with the quadratic term a bit later. The method for calculating it (or more correctly the primary part of the quadratic correction --see below) was standardized by the time the second edition of the Tables Requisite came out. That method is repeated in Norie's Table XXXV. And I'll just add again that Norie's "linear tables" were not an improvement in practicality. The calculation of the refraction between the two bodies (as one unit) could be easily tabulated without graphics. Those graphical "linear tables" were primarily created as a "copy protection" scheme, to put it in 1990s computer software terms. They may have also had a "cool" factor. As we know, navigators enjoy intriguing diagrams and graphical tools even today.

And you wrote:
"3) in the Thompson method, the "Table XVIII" give directly the correction of refraction combined with quadratic correction."

Yes.

There's a little detail that we haven't discussed yet. The quadratic correction, from a purely mathematical perspective, is really three terms. Any series expansion in two variables has two linear terms and three quadratic terms (see PS). In this case, we have the two linear terms proportional to the altitude corrections of the Moon and the other body. Then there's the largest quadratic term, which is proportional to the square of the Moon's altitude correction. This was the quadratic term first described and emphasized by Lyons, and it's what you would get from Norie's table xxxv. There is also a quadratic term proportional to the square of the altitude correction of the other body. That correction is always negligible except when the other body is below five degrees altitude. And those were always avoided in any case, so we can ignore that part of the quadratic correction. Finally, there is the "cross-term" of the quadratic correction which is proportional to the product of the Moon's altitude correction AND the other body's altitude correction. That additional quadratic correction was understood at least as early as the 1790s (see MyR's "Recherches" paper) but no one bothered to tabulate it until the 1820s. It was no big deal to add it, and it did not improve the accuracy of the computations more than about 6 seconds of arc. But it was a useful selling point for the newer tables. Finally it's interesting, and maybe a little amusing, that the later tables were also flawed in one way. The quadratic corrections were based on a mean value for the Moon's parallax. In fact, that was really their only flaw. The loss of accuracy from this defect was about the same as the improvement of accuracy from including the quadratic cross-term. ...One up. One down.

-FER
PS: Suppose I have something that depends on two variable, call them x and y. We can call that quantity the function f(x,y). It's a smoothly-varying function so I can calculate local variations in this quantity by doing a series expansion about some starting point x0, y0:
f(x,y) = f(x0,y0) + dx*A + dy*B +...
That is, the change in the function, to first order, is proportional to the small change in coordinates dx,dy away from x0,y0. We can do better by going to the next quadratic order in dx and dy. Since these are small offsets by supposition, their squares are very small. Generally:
f(x,y) = f(x0,y0) + dx*A + dy*B + dx*dx*C + 2*dx*dy*D + dy*dy*E +...
We've now got three quadratic terms. The factor of two in the cross-term is a useful convention. In terms of lunars, these offsets dy and dy and the altitude corrections for the Moon and Sun: dh_moon and dh_body. The factors A,B,C, etc. have to be calculated depending on the geometry of the specific altitudes and the observed lunar distance. But those quadratic bits are almost always very small, so we don't really need to calculate them every time. They go in a simple look-up table. So the series can be written:
f(x,y) = f(x0,y0) + dx*A + dy*B + Q.
We've already discussed the other simplifications and re-arrangements. There are various trig identities that let us calculate A in different ways, some more efficient but not by a big margin. There's also an interesting trick due to Legendre that lets us dispense with Q altogether. The catch is that you have to work up the linear terms twice. This trick was popular in some very short methods starting in the late 19th century, and it was the heart of the very last method which was published in the appendix of the British Nautical Almanac as late as 1919.

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