It seems that using SWR is a tradition (historical legacy). But even in the history of electronics I can't find why SWR was considered more useful compared with the reflection coefficient. What are/were the reasons to use SWR?
2 Answers
Because you could measure it with a slotted line - https://en.wikipedia.org/wiki/Slotted_line
As you slid the detector along the line, VSWR was simply the ratio of the peak amplitude to the trough amplitude. A slotted line is still something you can make with simple mechanical workshop equipment (well up to several GHz at least).
This is one of the things folk used before we had network analyzers.

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What are/were the reasons to use SWR?
SWR tells you directly what the maximum and minimum standing wave amplitudes are on a transmission line along the length of a line (example in red): -
Image from here.
Reflection coefficient directly tells you what the amplitude of the reflected wave will be; it's another step to find SWR. Reflection coefficient: -
Then to find SWR you use this.
In other words they calculate different things. However, they are easily interchangeable.
What are the cases when an electric engeneer needs to know maximum and minimum standing wave amplitudes and do not need to know the amplitude of the reflected wave?
Example
So, if you are designing a communication link between two points, you need to consider the SWR in case the amplitude of the signal present at one end or the other (or maybe the middle) is too low to be adequately received.

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Thank you, but it's not a complete answer on question "What are/were the reasons to use SWR?". I will clarify my question. What are the cases when an electric engeneer needs to know maximum and minimum standing wave amplitudes and do not need to know the amplitude of the reflected wave? – Arseniy Dec 21 '21 at 09:00
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I've added an example. – Andy aka Dec 21 '21 at 09:06
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Would any kind person tell me where the error is in my answer that attracted a downvote? I'm quite willing to fix it. – Andy aka Dec 21 '21 at 09:47
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The original question is "What are the reasons to use SWR instead of the reflection coefficient?". So the good answer must contains "The first reason is... the second reason is...". Instead, your answer contains description "the SWR is... the reflection coefficient is...". Actually you answered to enother question. I try to clarify my question and you add "you need to consider the SWR in case the amplitude ... is too low". This addition returns us to original question "What are the reasons to use SWR instead of the reflection coefficient?". [Edited by a moderator.] – Arseniy Dec 21 '21 at 10:15
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I added a specific case when you asked me (within minutes) @Arseniy and, in the body of your question you said this: `What are/were the reasons to use SWR?` specifically and I answered that specifically. If your title contains a question that's fine but, it's the body of your question where things need addressing. – Andy aka Dec 21 '21 at 10:22