Why should the description not be shown?
I think its by design that we do show it for every extension that is installed.

As you can see on the pictures, so you say it is normal behaviour for joomla ? Is there a way to prevent manifest's description from showing up in installation page ?

And is it possible for the plugin to echo some text only in the plugin's description page ?
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As you can see on the pictures, so you say it is normal behaviour for joomla ?

Yes, it is the expected behaviour.

Is there a way to prevent manifest's description from showing up in installation page ?

No, you can't hide it. The manifest description is meant to be shown there.
However you can show different descriptions during installation/upgrade and when editing the plugin. In the extension manager, the sys.ini language file is loaded, while when you edit the plugin it will load the regular language file. Thus if you remove the buttons and copyright from the sys.ini language file, you should get about what you want.

What you also can do is create a description fieldset in your manifest and put your extended description into that fieldset. When you edit the plugin, you will see the short description from the manifest and below a link to get an extended description, which basically selects the "description" tab.

What you also can do is create a description fieldset in your manifest and put your extended description into that fieldset.

Actually, the pictures I showed you are from another plugin. I have already a description fieldset (metadata) in my plugin's manifest. It is that one that is showing up after installation and in the plugin's editor. How can I do like you said : to have a link pointing to another description.
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How can I do like you said : to have a link pointing to another description ?
You may blame the J!Tracker Application for transmitting this comment.

As said, create a fieldset named "description" in your manifest and add some fields to it with the extended description you want to show. Joomla will automatically do the rest.

I'm going to close this issue as it's not a bug and issues are not meant for support requests.
Please open a thread at the Google Mailinglist: https://groups.google.com/forum/#!forum/joomla-dev-general