How to add Summary Links WP in page Layout in SP2013 Standard?

I’m coming to an issue :

1. I want to create a page layout whith a Summary Links Webpart.
2. I went to the design Manager
3. I chose my layout
4. I get the Snippet Gallery
5. I get the Summary Links WP code
6. I Copied/Pasted it to my layout via Sharepoint Designer
7. I checked the code : nothing says “only in display mode”.

When i go to a page using this layout, I can see the title of the WP. When i want to edit the page, and add links to the webpart, i can’t : i only see the title of the page, but nothing more, nor the “add group” nor “add links” buttons. So i can’t edit my WP and add Summary links…

And this is what i want : i want the users to add Links to this specific WP. I don’t want them to always be forced to add the WP when they create a page.

Do you have any idea on how to achieve this?

Thanks a lot for your help.
Gaëlle