I was able to solve. It was necessary to set the $incrementing property to false in the MemberLinkType model:

<?php

namespace App;

use Illuminate\Database\Eloquent\Model;

class MemberLinkType extends Model
{
    public $timestamps = false;    
    public $incrementing = false;    
    protected $primaryKey = 'name';    

}

After that, I got what I wanted:

$link = MemberLink::find(1)
=> App\MemberLink {#3071
     id: 1,
     member_link_type: "parent",
     from_member_id: 11,
     to_member_id: 12,
     created_at: "2018-08-14 15:03:22",
     updated_at: "2018-08-14 15:03:22",
     from: App\Member {#3065
       id: 11,
       name: "Gabriela Espinoza",           
     },
     to: App\Member {#3067
       id: 12,
       name: "Felipe Assunção",           
     },
     type: App\MemberLinkType {#3066
       name: "parent",
       category: "family",
       description: "Parent",
       description_on_male: "dad",
       description_on_female: "mom",
       reverse_link: "child",
     },
   }