This is how it is translated in to the suttas. For example, it SN12.2:
The four principal states, and form derived from the four principal states.
Cattāro ca mahābhūtā, catunnañca mahābhūtānaṁ upādāyarūpaṁ.
This is called form.
Idaṁ vuccati rūpaṁ
We also have SN22.79:
"Mendicants, an unlearned ordinary person might become disillusioned, dispassionate, and freed from this body made up of the four principal states.
"assutavā, bhikkhave, puthujjano imasmiṁ cātumahābhūtikasmiṁ kāyasmiṁ nibbindeyyapi virajjeyyapi vimucceyyapi
Why is that?
Taṁ kissa hetu?
This body made up of the four principal states is seen to accumulate and disperse, to be taken up and laid to rest.
Dissati, bhikkhave, imassa cātumahābhūtikassa kāyassa ācayopi apacayopi ādānampi nikkhepanampi.
That’s why an unlearned ordinary person might become disillusioned, dispassionate, and freed from it.
Tasmā tatrāssutavā puthujjano nibbindeyyapi virajjeyyapi vimucceyyapi.
It is worth noting that cātumahābhūtika is translated as composed of four primary states; made of the four great elements. I can understand how the fire element can function as a determinant of form. For example, in SN22.79 we read:
And why do you call it form?
Kiñca, bhikkhave, rūpaṁ vadetha?
It’s deformed; that’s why it’s called ‘form’.
Ruppatīti kho, bhikkhave, tasmā 'rūpan’ti vuccati.
Deformed by what?
Kena ruppati?
Deformed by cold, heat, hunger, and thirst, and deformed by the touch of flies, mosquitoes, wind, sun, and reptiles.
Sītenapi ruppati, uṇhenapi ruppati, jighacchāyapi ruppati, pipāsāyapi ruppati, ḍaṁsamakasavātātapasarīsapasamphassenapi ruppati.
Cold and heat can form water into ice, steam or liquid. As we are accustomed to water in the liquid state, it become a principle of liquidity. Fire in that sense is a determinant rather than a principle.
The implications of above might explain why the fire element is treated differently. In anthropology, discovering fire is considered a turning point in human development. In mythology, it was given to humans by a rebellion god. In religion, taking it as a principle state is associated with pain (niraya) but using it as a tool can generate warmth and light. In behaviorism and psychology, humans are the only known species that smokes, which indicates the human ability to think and behave in symbolic ways.
In MN43, it is used to differeniate a dead person from someone who attains the cessation of perception and feelings:
“What’s the difference between someone who has passed away and a mendicant who has attained the cessation of perception and feeling?”
"Yvāyaṁ, āvuso, mato kālaṅkato, yo cāyaṁ bhikkhu saññāvedayitanirodhaṁ samāpanno—imesaṁ kiṁ nānākaraṇan"ti?
"When someone dies, their physical, verbal, and mental processes have ceased and stilled; their vitality is spent; their warmth is dissipated; and their faculties have disintegrated.
"Yvāyaṁ, āvuso, mato kālaṅkato tassa kāyasaṅkhārā niruddhā paṭippassaddhā, vacīsaṅkhārā niruddhā paṭippassaddhā, cittasaṅkhārā niruddhā paṭippassaddhā, āyu parikkhīṇo, usmā vūpasantā, indriyāni paribhinnāni.
When a mendicant has attained the cessation of perception and feeling, their physical, verbal, and mental processes have ceased and stilled. But their vitality is not spent; their warmth is not dissipated; and their faculties are very clear.
Yo cāyaṁ bhikkhu saññāvedayitanirodhaṁ samāpanno tassapi kāyasaṅkhārā niruddhā paṭippassaddhā, vacīsaṅkhārā niruddhā paṭippassaddhā, cittasaṅkhārā niruddhā paṭippassaddhā, āyu na parikkhīṇo, usmā avūpasantā, indriyāni vippasannāni.
The above made me wonder what makes the fire a primary element or principle as per my explanation in the OP.
