The way I do it is 70% of how you described in your Edit.
When the user enters their save game name, have another save file called ‘SaveGameList’ or something, with a variable called ‘SaveGameName’. Then once your user enters their save game name, it saves this to the save file you already have, but then also saves the game name to the SaveGameList (your user doesnt need to enter this twice, you can jsut send the same name twice).
To Load the list, load your SaveGameList and pull the SaveGameName array from the file, each of the entries in this List will relate to one of your saves.
Hope this helps