I mean that's the physical reason that's happening (or menthol etc is tricking them), but we call them cold recepting because relative to base temperature, they (the neurons) fire more when it's colder compared to heat sensing which fire more with heat (or spicy like capsaicin and some other compounds). I'm actually unsure (fairly confident there is) if there is multiple subtypes of each.

edit: just noticed the other reply was to yourself, sorry!