Kāyika vedanā are just simple sensations, as in the Sakalika sutta:
Tena kho pana samayena bhagavato pādo sakalikāya khato hoti, bhusā sudaṃ bhagavato vedanā vattanti sārīrikā vedanā dukkhā tibbā kharā kaṭukā asātā amanāpā. Tā sudaṃ bhagavā sato sampajāno adhivāseti avihaññamāno.
Now at that time his foot had been pierced by a stone sliver. Excruciating were the bodily sensations that developed within him — painful, fierce, sharp, wracking, repellent, disagreeable — but he endured them mindful, alert, & unperturbed.
or the Sivaka sutta:
“Master Gotama, there are some priests & contemplatives who are of this doctrine, this view: Whatever an individual senses — pleasure, pain, neither-pleasure-nor-pain — is entirely caused by what was done before. Now what does Master Gotama say to that?”
[The Buddha:] “There are cases where some sensations arise based on bile. You yourself should know how some sensations arise based on bile. Even the world is agreed on how some sensations arise based on bile. So any priests & contemplatives who are of the doctrine & view that whatever an individual senses — pleasure, pain, neither-pleasure-nor-pain — is entirely caused by what was done before — slip past what they themselves know, slip past what is agreed on by the world. Therefore I say that those priests & contemplatives are wrong.”
“There are cases where some sensations arise based on phlegm… based on internal winds… based on a combination of bodily humors… from the change of the seasons… from uneven care of the body… from harsh treatment… from the result of kamma. You yourself should know how some sensations arise from the result of kamma. Even the world is agreed on how some sensations arise from the result of kamma. So any priests & contemplatives who are of the doctrine & view that whatever an individual senses — pleasure, pain, neither pleasure-nor-pain — is entirely caused by what was done before — slip past what they themselves know, slip past what is agreed on by the world. Therefore I say that those priests & contemplatives are wrong.”
A survey of the use of the term vedanā (“sensations”) in the Pali Nikayas
Cetasikā vedanā , - those that arise due to the contact of ‘mano’:
394. "Katama~nca, bhikkhave, domanassa.m? Ya.m kho, bhikkhave, cetasika.m dukkha.m cetasika.m asaata.m manosamphassaja.m dukkha.m asaata.m vedayita.m, ida.m vuccati, bhikkhave, domanassa.m.
(DN 22, Mahasatipatthana sutta)
for example, when one recalls acquisition or non-acquisition:
Tattha katamāni cha gehasitāni somanassāni: cakkhuviññeyyānaṃ rūpānaṃ iṭṭhānaṃ kantānaṃ manāpānaṃ manoramānaṃ lokāmisapaṭisaṃyuttānaṃ paṭilābhaṃ vā paṭilābhato samanupassato pubbe vā paṭiladdhapubbaṃ atītaṃ niruddhaṃ vipariṇataṃ samanussarato uppajjati somanassaṃ, yaṃ evarūpaṃ somanassaṃ idaṃ vuccati gehasitaṃ somanassaṃ.
"And what are the six kinds of household joy (somanassa)? The joy that arises when one regards as an acquisition the acquisition of forms cognizable by the eye — agreeable, pleasing, charming, endearing, connected with worldly baits — or when one recalls the previous acquisition of such forms after they have passed, ceased, & changed: That is called household joy. (Similarly with sounds, smells, tastes, tactile sensations, & ideas.)
Tattha katamāni cha gehasitāni domanassāni: cakkhuviññeyyānaṃ rūpānaṃ iṭṭhānaṃ kantānaṃ manāpānaṃ manoramānaṃ lokāmisapaṭisaṃyuttānaṃ appaṭilābhaṃ vā appaṭilābhato samanupassato pubbe vā appaṭiladdhapubbaṃ atītaṃ niruddhaṃ vipariṇataṃ samanussarato uppajjati domanassaṃ.
"And what are the six kinds of household distress (domanassa)? The distress that arises when one regards as a non-acquisition the non-acquisition of forms cognizable by the eye — agreeable, pleasing, charming, endearing, connected with worldly baits — or when one recalls the previous non-acquisition of such forms after they have passed, ceased, & changed: That is called household distress. (Similarly with sounds, smells, tastes, tactile sensations, & ideas.)
Salayatana-vibhanga sutta http://www.accesstoinsight.org/tipitaka/mn/mn.137.than.html
indeed arise not due to the external stimulation of sense organs, if we understand sense organs in their Western definition. Yet in ancient Indian worldview, ‘mano’ belongs to sense faculties.
In the case of four jhanas the highest enjoyment fills the body:
Seyyathāpi bhikkhave uppaliniyaṃ vā paduminiyaṃ vā puṇḍarīkiniyaṃ vā appekaccāni uppalāni vā padumāni vā puṇḍarīkāni vā udake jātāni udake saṃvaddhāni udakānuggatāni anto nimuggaposīni, tāni yāvaggā yāva mūlā sītena vārinā abhisannāni parisannāni paripūrāni paripphuṭāni. Nāssa kiñci sabbāvataṃ uppalānaṃ vā padumānaṃ vā puṇḍarīkānaṃ vā sītena vārinā apphuṭaṃ assa. Evameva kho bhikkhave bhikkhu imameva kāyaṃ nippītikena sukhena abhisandeti parisandeti paripūreti parippharati. Nāssa kiñci sabbāvato kāyassa nippītikena sukhena apphuṭaṃ hoti. Ariyassa bhikkhave pañcaṅgikassa sammāsamādhissa ayaṃ tatiyā bhāvanā.
"Just as in a blue-, white-, or red-lotus pond, there may be some of the blue, white, or red lotuses which, born and growing in the water, stay immersed in the water and flourish without standing up out of the water, so that they are permeated and pervaded, suffused and filled with cool water from their roots to their tips, and nothing of those blue, white, or red lotuses would be unpervaded with cool water; even so, the monk permeates and pervades, suffuses and fills this very body with the pleasure divested of rapture. There is nothing of his entire body unpervaded with pleasure divested of rapture. This is the third development of the five-factored noble right concentration.
(4) Puna ca paraṃ bhikkhave bhikkhu sukhassa ca pāhāṇā dukkhassa ca pahāṇā pubbeva somanassadomanassānaṃ atthaṅgamā adukkhaṃ asukhaṃ upekkhāsatipārisuddhiṃ catutthaṃ jhānaṃ upasampajja viharati.
"And furthermore, with the abandoning of pleasure and stress — as with the earlier disappearance of elation and distress — he enters and remains in the fourth jhana: purity of equanimity and mindfulness, neither-pleasure-nor-pain.
In Salla sutta two kinds of vedanā are placed alongside, as similar phenomena:
assutavā, bhikkhave, puthujjano dukkhāya vedanāya phuṭṭho samāno socati kilamati paridevati urattāḷiṃ kandati sammohaṃ āpajjati. So dve vedanā vedayati—kāyikañca, cetasikañca. Seyyathāpi, bhikkhave, purisaṃ sallena vijjheyya. Tamenaṃ dutiyena sallena anuvedhaṃ vijjheyya. Evañhi so, bhikkhave, puriso dvisallena vedanaṃ vedayati. Evameva kho, bhikkhave, assutavā puthujjano dukkhāya vedanāya phuṭṭho samāno socati kilamati paridevati urattāḷiṃ kandati sammohaṃ āpajjati. So dve vedanā vedayati—kāyikañca, cetasikañca.
uninstructed worldling is being contacted by a painful sensation, he sorrows, grieves, and laments; he weeps beating his breast and becomes distraught. He feels two sensations—a bodily one and a mental one. Suppose they were to strike a man with a dart, and then they would strike him immediately afterwards with a second dart, so that the man would feel a feeling caused by two darts. So too, when the uninstructed worldling is being contacted by a painful feeling … he feels two feelings—a bodily one and a mental one.
So evidently the heightened introspection ability of ancient Indian Ariyas, and their ability to induce intense experiences not related to input from bodily senses, helped them to notice and categorize vedanā that arise due to contact of mano and are felt in the body similarly to vedanā that arise due to contact of five sense organs .
Seems like there’s no exact equivalent in English, however, the term ‘sensation’ allows some space for sensations that arise due to contact of ‘mano’:
Sensation, 1 … a) a feeling, especially a strange one, caused by a particular experience
He had the uncomfortable sensation that he was being watched.
and helps to distinguish vedanā from emotions.