This question is about words. The Pali words are 'vedana' ('feelings') & 'sankhara' ('formations').
Happy, sad, elated, down - more like an emotion = sankhara ('formations'). To quote:
When one abides inflamed by lust, fettered, infatuated, contemplating gratification, then the five aggregates affected by clinging are built
up for oneself in the future; and one’s craving—which brings renewal
of being, is accompanied by delight and lust, and delights in this and
that—increases. One’s bodily and mental troubles increase, one’s
bodily and mental torments increase, one’s bodily and mental fevers
increase and one experiences bodily and mental suffering (kāyadukkhampi cetodukkhampi paṭisaṃvedeti). MN 149
Hot, cold, tingly, numb, all kinds of body pains or pleasant sensations = 'vedana'. To quote:
And why, bhikkhus, do you call it feeling (vedanaṃ)? ‘It feels (vedayatī),’ bhikkhus, therefore it is called feeling. And what does it feel? It
feels pleasure, it feels pain, it feels neither-pain-nor-pleasure. ‘It
feels,’ bhikkhus, therefore it is called feeling. SN 22.79
But some body pains are formations, such as when due to anger, stress, anxiety, etc. To quote:
When one abides inflamed by lust, fettered, infatuated, contemplating gratification, then the five aggregates affected by clinging are built
up for oneself in the future; and one’s craving—which brings renewal
of being, is accompanied by delight and lust, and delights in this and
that—increases. One’s bodily and mental troubles increase, one’s
bodily and mental torments increase, one’s bodily and mental fevers
increase and one experiences bodily and mental suffering (kāyadukkhampi cetodukkhampi paṭisaṃvedeti). MN 149
The core feature of 'vedana' ('feelings' or 'sensations') is they arise from sense contact. To quote:
‘The six classes of feeling should be understood.’ So it was said. And with reference to what was this said? Dependent on the eye and
forms, eye-consciousness arises; the meeting of the three is contact;
with contact as condition there is feeling. Dependent on the ear and
sounds, ear-consciousness arises; the meeting of the three is contact;
with contact as condition there is feeling. Dependent on the nose and
odours, nose-consciousness arises; the meeting of the three is
contact; with contact as condition there is feeling. Dependent on the
tongue and flavours, tongue-consciousness arises; the meeting of the
three is contact; with contact as condition there is feeling.
Dependent on the body and tangibles, body-consciousness arises; the
meeting of the three is contact; with contact as condition there is
feeling. Dependent on the mind and mind-objects, mind-consciousness
arises; the meeting of the three is contact; with contact as condition
there is feeling. So it was with reference to this that it was said:
‘The six classes of feeling should be understood. ’ This is the fifth
set of six. MN 148
Where as formations arise from defilement (craving) & thinking (sankhara). To quote:
What one feels, one perceives (labels in the mind). What one perceives, one thinks about. What one thinks about, one objectifies.
Based on what a person objectifies, the perceptions & categories of
objectification assail him/her with regard to past, present & future
forms cognizable via the eye. MN 18
