I'll divide my answer in two cases. First, I'll talk about **liquids in motion** (assuming incompressible flow). Then, I'll talk about **liquids at rest**.
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Liquid Flow:
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Reading the comments of [this](https://youtu.be/_VDVi_zq3nk) YouTube video about piezometers made by Donald Elger, I found the answer for this case:
>Why is it [the pressure measurement with piezometer] taken from the middle of the pipe?
>
>Elger's answer: The pressure variation across a section of a pipe is hydrostatic; thus, the pressure will vary linearly with radius and the pressure at the center of the pipe is the average pressure. If you use this value of pressure in your calculations, this will be give you the **most accurate results**. Thus, engineers nearly always apply or measure the pressure at the center of the pipe.
The question that came to me as soon as I read this was: ["Why using average pressure in calculations gives the most accurate results?"](https://physics.stackexchange.com/questions/437410).
(*Note: I recommend reading my answer to this question before proceeding*)
Briefly, in general, the average pressure gives the most accurate results if used in calculations because there are many applications/cases in which the locations with $P=P_{average}$ are the best places for experimental data collection.
In the case of a pipe, this location is its centerline. So, I believe that this is why textbooks generally choose this location in case of liquids in motion: the centerline is associated with $P_{average}$ that, in its turn, is associated with the best places for experimental data collection for many applications.
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Liquids at rest:
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For this case, firstly I would like to quote part of the answer written by David White to my question ["Where is the right place to put the pressure gauge to measure the pressure of a tank?"](https://physics.stackexchange.com/a/436482/102936):
>The location depends on why you are measuring the pressure. There will be a process reason for the pressure measurement, and that will determine the location of the pressure measuring device.
When textbooks present pressure gauges/manometers/piezometers for the first time, the presentation is usually "application neutral" (i.e., there's no process reason), the diagrams/sketches/figures are only to illustrate the concepts/formulas. Therefore there are no best points as in the liquid flow case, for two reasons:
- There is no process reason that determines the location of measurement;
- Since the liquid is at rest, there are no points that lead to most accurate results, they all provide the same accuracy.
**But the authors need to choose a point to do the pressure-related calculations...**
After everything I've researched, my hypothesis is that the "point choice" of hydrostatics was imported from hydrodynamics. So, instead of choosing a random point to pressure-related calculations, they choose one that at least has importance/meaning for other areas of Fluid Mechanics.