Rig Veda 7:59:12 and Yajur Veda 3.60 are called Mahā mṛtyuñjaya mantra Mantra. If we divide the word Tryambaka, it is Tri+ambaka. त्र्यंबक = त्रि + अम्बक. The two words are joined based on यणादेश संधि (Yaṇādeśa sandhi).
The mantra is:
ॐ त्र्यम्बकं यजामहे सुगन्धिं पुष्टिवर्धनम् |
उर्वारुकमिव बन्धनान्मृत्योर्मुक्षीय माऽमृतात् ||
Oṃ tryambakaṃ yajāmahe sugandhiṃ puṣṭivardhanam |
urvārukamiva bandhanānmṛtyormukṣīya mā'mṛtāt ||
Meanings:
1. Om, We Worship the Three-Eyed One (Lord Shiva), Who is Fragrant (Spiritual Essence) and Who Nourishes all beings. May He severe our Bondage of Samsara (Worldly Life), like a Cucumber (severed from the bondage of its Creeper), and thus Liberate us from the Fear of Death, by making us realize that we are never separated from our Immortal Nature.
2. Om, We Worship the Three-Eyed One (Lord Shiva), Who is Fragrant (Spiritual Essence) and Who Nourishes all beings. May He severe our Bondage of Samsara (Worldly Life), like a Cucumber (severed from the bondage of its Creeper), and thus Liberate us from the Fear of Death, Give me some life rejuvenating nectar.
3. Om, We Worship the Three-Eyed One (Lord Shiva), Who is Fragrant (Spiritual Essence) and Who Nourishes all beings. May He sever our Bondage of Samsara (Worldly Life), like a Cucumber (severed from the bondage of its Creeper), and thus Liberate us from the Fear of Death, Free me from certain death.
Differences in the translations is due to the different interpretation of "माऽमृतात्" at the end of the mantra.
Ambaka means an eye in Sanskrit whereas Ambika is an epithet of a mother. Ambika is also a character in Mahabharata who is the mother of Dhritarashtra. Ambaka and Ambika are different words.
Lord Shiva is known to have three eyes. Hence, Tryambaka means the one who has three eyes.
On a side note, The consort of Shiva, Lalita Devi is also known as Tryambakā.
From Lalitā sahasra nāma,
trivarga dhātrī śubhagā tryaṃbakā triguṇātmikā
She who gives dharma, Assets and pleasure, She who is pleasing to look at
She who has three eyes, She who is personification of three gunas viz Thamo, Rajo and Sattva