RE: Ok. How can I be sure that they modify only THEIR entry?
I was speaking in context of access to WP in general. When Link is configured per the instructions, the user will be able to edit their directory entry.
RE: How can I allow them to modify all the fields?
They will be able to edit their entire entry.
Based on your questions, it seems like you just want a user to be able to add and edit their own entry. To do this, configure Link per the instructions. Do not enable the “Submit Entry” link option.
Set up a process, using either the core WP user registration process or a plugin which processes user registration. After the user is registered, they will be able to add/edit their directory entry. I suggest setting users as Subscribers. With that role, the will only be able to edit their name and password as well as maintain their directory entry per the Link configuration.
Hope this better answers your questions.