Fresh peppercorns certainly retain their heat during cooking. They are occasionally used in Thai stir-fries. If you eat enough, you will get crazy effects 4-12 hours later: The palms of your hands and bottom of your feet will tingle. It's wild.

Also, chillis lose some heat with cooking. Again: From Thai cooking, fresh chillis are used sparingly as a topping, but Bird's Eye chillis can be used in a stir fry enough to kill a small child (joke, but they can use a lot).

English has so many words, and yet it doesn't have a separate word for the spicy hotness that cannot be confused with heat?

It's funny; the ambiguity of English "hot" makes it difficult to even think clearly about the different kinds of hotness. Ginger and garlic can produce similar sensations of hotness to what you get with chilli; I sometimes make a Nepalese-style lamb "curry" that has no chilli or pepper in it. It's very mild, but you could be forgiven for thinking it's flavoured with chilli.

It seems to be because the sensation induced in the mouth is literally that of heat. I remember Adam Ragusea on his excellent YouTube channel pointing out that capsaicin makes the mouth much more sensitive to temperature. So funnily enough, the sensation of something being too hot temperature-wise, and something being spicy is pretty much the same thing. It's also why cold water seems to help for a bit, but does nothing to eliminate the effects of the capsaicin as opposed to something like milk.

Perhaps that's an example where the word you use affects the experience you feel.

In Italian it's "piccante" which comes from archaic "piccare" "to sting". And to me, chili pepper sting and punch more than give me a feeling of heat

There's 'spicy', but then it can also be ambiguous (unless your tolerance is really low) with something that is full of flavourful spice notes but without the, er, heat.

I'm not sure Hindi does either though? Garam is certainly ambiguous.