No, independent would mean you could cease that source of income and maintain your lifestyle. If you save 50% of your first paycheck and then quit I doubt that would be the case.
Being able to set that much aside would definitely make one wealthy (or live a very austere lifestyle) and fast track them toward independence, but it’s not an automatic qualifier.
Oh nice, I just replayed The Last Autumn since the trees are changing color, just waiting on the first snowfall to play the main game. Let me know if there’s anything especially challenging you want advice on!