Most of the Tibetan Buddhism Schools have teachings about a svabhavakâya or svabhavikakâya, named either the third or fourth kaya, sometimes described as the sum of the other ones, sometimes as the basis for the other ones.
1) Nyingma School
From « The Wish-Fulfilling Jewel : The Practice of Guru Yoga According to the Longchen Nyingthig Tradition (Rigdzin Jigme Lingpa)» :
« Receiving the Four Enpowerments – This fourth or word initiation is the introduction to the natural state of all phenomena ; through it we become a proper vessel for the practice of Dzogchen, the Great Perfection…It is the ultimate buddhahood, the indivisibility of the three kayas, or the svabhavikakaya, the body of the true nature.
2) Kagyu Schools
From « Mahamudra and related instructions – Core Teachings of the Kagyu Schools» :
The Svabhavakaya : This is great peace and is the nature of all phenomena. It is attained through the power of the dharmakaya, through realisation. The vajrayana calls this the body of great bliss (mahâsukhakâya) because its distinctive quality is supreme, unchanging bliss. Ârya Nâgârjuna has said : « I pay homage to that which is free from the activity of the three realms ; which is the equality of space ; which is the nature of all things ;… Praise to the Three Kâyas (Kayâtrayastotra), Toh 1123, Tengyur, bstod tshogs,ka,70b3.
Other references to the svabhavakâya (from the same book) :
« The svabhavakâya is the dharmakaya of the tathâgatas, because it is the locus of power over everything. » Asanga – Mâhâyanasamgraha, Toh 4048, Tengyur, sems tsam, ri, 37a4. The Tibetan adds the word « phenomena » to make « power over all phenomena »
« The categories of the kâyas of the buddhas : There is the svabhava, the sambogha, and the other kâya is the nirmâna. The first is the basis for the other two. » Sûtrâlamkâra 10:60,11a7.
« The svabhavakâya is equal and subtle. » Sûtrâlamkâra 10:62,11b1. The Dergé Tengyur has « Rang bzhin sku ni mnyam pa dang »
« The first kâya (svabhavakaya) has the qualities of liberation, such as the powers and so on ». Sublime Continuum, 3:2,65b2
From Jamgön Kongtrul – The Treasury of Knowledge :
Talking about the results of practice : « The uncommon transformation is that the physical channels transform into the nirmanakaya, the channel syllabes into sambhogakaya, the constituent elixir into dharmakaya and great bliss, and the core energy current of pristine awareness transform into the svabhavikakâya. »
« … svabhavikakâya is characterized as emptiness, which is to say, the nature of all phenomena, a nature that is free of all elaboration and completely pure ; »
« There are four kayas when one adds the svabhâvikakâya (enlightened dimension of the very essence of being itself) of innate presence, or mahâsukhâya, to the three kayas.
3) Sakya Schools
From the Vajra Lines of the Path with the Results (Virupa) – Explication of the Treatise for Nyak :
« The naturally spontaneous, utterly pure svâbhâvikakâya essence body is achieved. The result is perfected. »
« The fourth initiation dissolves the pulsations of the vital winds…The vital winds are transformed and ‘omnipresent enlightened body, speech, and mind, the svâbhâvikakâya essence body’, is actualized. »
These descriptions , root texts and commentaries are supporting the idea highlighted about the principle of svabhava in Mahayana Buddhism, as not relative to a permanent quality, but rather as an essence. It is often described as the nature of the phenomena, as a basis, never as eternal.
The svabhāva-kāya or svābhāvika-kāya is indeed an important topic for us in relation to the svabhāva teachings of the Book of Dzyan because, as shown by the quotations you posted, it is accepted as a reality by every school of Tibetan Buddhism. It seems that the basic definition of it used by them comes from Maitreya’s Abhisamayālaṃkāra, chapter 8, verse 1. I understand this verse as: “The svābhāvika body of the sage [the buddha] is defined as the nature (prakṛti) of those uncontaminated (nirāsrava) dharmas which have attained all-aspect (sarvākāra) purity.” The earliest extant commentary, that by Ārya Vimuktisena, takes the svābhāvika-kāya as a synonym of the dharma-kāya, and thus accepts three buddha-bodies. The famous Indian commentator Haribhadra takes the svābhāvika-kāya as distinct from the dharma-kāya, and thus accepts four buddha-bodies. The Gelugpa teacher Tsongkhapa followed Haribhadra in accepting four buddha-bodies, while the Sakyapa teacher Gorampa followed Ārya Vimuktisena in accepting three buddha-bodies (see: Buddhahood Embodied: Sources of Controversy in India and Tibet, by John J. Makransky, SUNY Press, 1997). The Jonangpa teacher Dolpopa, who preceded both of these other Tibetan teachers, accepted either three or four (Mountain Doctrine, pp. 428-429). Four are accepted in Kālacakra in order to correspond with the four states (waking, dreaming, deep sleep, and the fourth), etc.