The point in time when one becomes a Buddhist is the time when one takes refuge in the Triple Gems, i.e. the Buddha, Dharma and Sangha. This is turning the mind towards the Buddhist path, and be protected by the refuge of the Triple Gems.
Here is a traditional prayer for taking refuge in the Triple Gem, commonly recited in Theravāda Buddhist practice:
Palms together, in reverence to the Buddha, Dhamma, and Sangha.
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Pali Version:
Buddhaṃ saraṇaṃ gacchāmi
Dhammaṃ saraṇaṃ gacchāmi
Saṅghaṃ saraṇaṃ gacchāmi
Dutiyampi Buddhaṃ saraṇaṃ gacchāmi
Dutiyampi Dhammaṃ saraṇaṃ gacchāmi
Dutiyampi Saṅghaṃ saraṇaṃ gacchāmi
Tatiyampi Buddhaṃ saraṇaṃ gacchāmi
Tatiyampi Dhammaṃ saraṇaṃ gacchāmi
Tatiyampi Saṅghaṃ saraṇaṃ gacchāmi
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English Translation:
I go to the Buddha for refuge.
I go to the Dhamma for refuge.
I go to the Sangha for refuge.
For the second time, I go to the Buddha for refuge.
For the second time, I go to the Dhamma for refuge.
For the second time, I go to the Sangha for refuge.
For the third time, I go to the Buddha for refuge.
For the third time, I go to the Dhamma for refuge.
For the third time, I go to the Sangha for refuge.
This is an essential practice one can do everyday. It renews one’s commitment towards full and complete awakening.