Hi
My husband had nut allergies so I know this first hand.
A peanut is a legume, like a bean. Not a nut. Peanuts grow underground.
A nut grows on a tree, examples of nuts are: almonds, cashews, walnuts, pecans....
Both peanuts and TREE nuts cause allergic reations in some people. They are different but both known allergens.
Some people are allergic only to peanuts, not TREE nuts. Some both. Some only the tree nuts. My husband can eat peanuts but will go into shock if he eats TREE nuts or coconuts.
Some people with a TREE NUT allergy are also allergic to coconuts (which are the seed of the coconut tree) also nutmeg is not 100% nutfree either. Nether is pesto.
FYI
It is best to treat all nuts with caution untill you are sure your lo isnt allergic.
As for seeds I am not sure, so I cant give advice. But I havent heard much about seed allergies. So pumpkin or sunflower may be introduced... A seed is much different then a nut. A seed is found in a fruit.