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Should I be apply more pressure on the green part rather than the red?
You're not supposed to apply pressure on the red area, the only possibility of doing that is tensing the muscle which pulls your tongue backward, which will make you unable to breath.
Honestly, I would not apply pressure anywhere. You can lightly apply pressure with your tongue in various areas. One or two spots may feel better to you. That is fine to rest your tongue there. Your body is your feedback signal.