It's hard to tell what's happening without seeing the entire project. The function on your screenshot should definitely create "score" key. If you previously parsed another data with "score_table", then the JSON should contain both keys:
{"score_table":{"id":1,"position_x":"0","position_y":"0","boss_counter":"0"},
"score":{"id":1,"position_x":"0","position_y":"0","boss_counter":"0"}}
If this doesn't happen, maybe the function doesn't work? Or another AJAX request immediately overwrites JSON with the old data from the server? I can only guess.
Learn to use the debugger tool (Shift+F4). Temporarily disable the loading function and test that the saving function works - check the contents of JSON in the debugger.