The botanical definition of nut and the common culinary definition of nut are different, and each is valid in its own sphere of usage:
True (botanical) nuts:
Beech (Fagus)
Chestnut (Castanea)
Oak (Quercus)
Stone-oak (Lithocarpus)
Tanoak (Notholithocarpus)
Family Betulaceae
Hazel, Filbert (Corylus)
Hornbeam (Carpinus)
Seeds which are not botanical nuts, but are commonly called so:
Almonds (edible seeds of drupe fruits)
Brazil nut (seed from a capsule)
Candlenut
Cashew (seed of an accessory fruit)
Chilean hazelnut or Gevuina
Macadamia (kernel of a follicle type fruit)
Malabar chestnut
Pecan (seed of a drupe fruit)
Mongongo
Peanut (seed from a species of legume)
Pine nut (seed of several species of pine)
Pistachio (seed of a thin-shelled drupe)
Walnut (Juglans)
As you see, rather few nuts are even edible! But in my experience, people with an allergy to one seed (peanut, especially) are appropriately cautious about approaching any other seed, whether called a nut or not.