Had problems trying to get this myself; it would appear to count each note but not give the achievement at 10. Don't know how common this problem is but I saw several threads of people expressing their frustration online. Decided to poke around the files a bit and found a solution.

I did not test starting at partial completion, you may need to get all 10 notes again but it will give the achievement this time. Worked for me on first attempt testing the mod. Only needs to be installed until the achievement is granted, then it can be removed.

Install instructions:
Merge the folder \data\ with Data0.pak (not Data3.pak) in \steamapps\common\Dying Light\DW. Always make backups first! There's a video on the mod page for more detailed install instructions.

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[Patch notes]

Changed (in achievements.scr)

Event("Evn_ItemTakenDropped")
        {
            ItemsSet("Collectable_Note_Hellraid_01;Collectable_Note_Hellraid_02;Collectable_Note_Hellraid_03;Collectable_Note_Hellraid_04;Collectable_Note_Hellraid_05;Collectable_Note_Hellraid_06;Collectable_Note_Hellraid_07;Collectable_Note_Hellraid_08;Collectable_Note_Hellraid_09;Collectable_Note_Hellraid_10");            
            AllDifferent(10);
            SaveToProfile();
            ForceCheckProgressForVer130();
        }

to

Event("Evn_ItemTakenDropped")
        {
            ItemsSet("Collectable_Note_Hellraid_01;Collectable_Note_Hellraid_02;Collectable_Note_Hellraid_03;Collectable_Note_Hellraid_04;Collectable_Note_Hellraid_05;Collectable_Note_Hellraid_06;Collectable_Note_Hellraid_07;Collectable_Note_Hellraid_08;Collectable_Note_Hellraid_09;Collectable_Note_Hellraid_10");            
        }

The problem exists somewhere in these 3 lines but I couldn't find out exactly what their code does:

AllDifferent(10);
SaveToProfile();
ForceCheckProgressForVer130();

AllDifferent and SaveToProfile are used elsewhere without issue so I guess ForceCheckProgressForVer130 is the culprit. No way to test this as I have the achievement now. To be safe I still left out the other 2 lines in case they have some weird interaction with that achievement specifically.