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The 9's and 10's I date cringe at "I'm sorry". They just see it as weak behavior.
But, point taken everywhere else.
Look, you can do whatever you want, but consider this:
If you let the idea of coming across as weak influence your behavior, that's exactly weak behavior.
Being supplicant is not the only form of neediness. Overcompensating the bad-boy persona is just as bad. Granted it will get you laid by certain (and damaged) women, but even those will eventually see through your masquerade. And except a festering pool of insecurities, there won't be much to see.
Life isn't black and white OP. And women will never judge an action in isolation. Actions are contrasted by the man behind them and by his motives.
If you apologize because you're scared shitless you'll never get your dick wet by anyone close to her level of hot, then yeah, she will cringe. That's a desperate man with a selfish motive.
If you apologize in an authentic fashion because you crossed a line, there's no ulterior motive. That's a quality guy just owning his mistake. He likes the girl enough to care about hurt feelings, but his life doesn't depend on it. That's anything but weak.