@kuruman, I agree with your earlier point directed at me in this thread that Pascal's principle is not what the OP was asking about -- the U-shaped tube is open, whereas the hydraulic jack is closed.
I also agree that the wording in the text could be confusing, especially the inclarity of the use of the term "height". When there are 2 liquids of different densities at the 2 ends of the tube, with both ends open to the atmosphere, as in the OP's image, you get variance in height of the liquids below the ends of the tube, similarly to, but in this instance more clearly labeled, as in the following image (from an illustration for a fluid statics problem video explanation presented
here):
It's a pretty good, reasonably on-topic (I think) video, so I'll leave it linked as a video, but I'll also go ahead and synopsize the diagram:
That diagram depicts water, which is of density (as specific gravity) 1, in one end of the tube, and mercury, of density about 13.6, in the other end. With both ends of the tube open to the atmosphere, the water end will be visibly higher above the equal pressure line in the tube, for equal weights of water and mercury above that line, the zero line.
In the diagram, the label says P1 = P2, which makes the line between them horizontal, and makes that line the zero line.
The zero line is set at the lower of the 2 mercury levels, because there's only mercury below that level, therefore only one fluid density, so the heights at that level being the same, the pressures are the same, just as we see equal liquid height in a U-shaped tube when there's only 1 liquid and both ends are open to the atmosphere.
If the water height above P1 is 15 cm, then the mercury height above P2 should be about 1/13.6th of that, or about 1.1 cm (the diagram's not drawn to scale), because mercury is 13.6 times as dense as water. The video explains that calculation in detail. (If instead of using mercury for a barometer you made a water barometer, then instead of atmospheric pressure being about 76.6 cm of Hg, you'd have about 10.4 m. of H2O.) The diagram poses the question what is ##\Delta h## equal to (I'd have called it h1-h2, instead of h2-h1, as on the diagram label); it is the (absolute) difference between the heights above zero of the water and the mercury, meaning 15 cm - 1.1 cm, which is 13.9 cm.
For a single fluid of uniform density, as in the 2 diagrams of pairs of tubes below, you get different heights at the 2 ends of the tube only if there's a difference between the pressures at the 2 ends.
Please note the difference between the use of the term
h in the 1st image vs
(H) in the second image.
In second of the pair of tubes in that image, P2 > P1, and
h refers to how far above the zero line the fluid at the lower-pressure end is.
In that image, (H) refers to how far zero is above the lower fluid level,
plus how far the higher level is above zero.
One of the
(endearing?) bewildering qualities of standard definitions is that there are so many of them to choose from.