I told Korean friends not to label a (non-commercial) package of seaweed as "seafood", but it is from the sea and it is food, so now I'm not sure.
How common is it to refer to "seaweed" as "seafood"?
Does this vary from place to place?
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I wasn't able to find any dictionary which included plant-based food in the definition of "seafood."
Merriam-Webster (American English):
edible marine fish and shellfish
Edible fish or shellfish from the sea.
Shellfish and other edible marine fish.
Some other sources:
So it seems like seaweed can be considered seafood, but it does not seem to be common practice. I would never use the term to refer to seaweed or other plant-based food.
A state of Connecticut definition (for the purpose of a specific law concerning the requirement of a certificate of registration for the retail sale of seafood) is:
"Seafood" means all fresh or saltwater finfish, molluscan shellfish, crustaceans and other forms of aquatic animal life
which would lead us to believe that plant products are not included in the category 'seafood'.
but it is from the sea and it is food, so now I'm not sure
This is a common misconception. Language categories don't work at all like mathematical sets, but most of us are unaware of it. So people try to treat them as they were working that way, or even start insisting that language is "incorrect" because it doesn't obey the same rules.
In mathematics, sets are defined by their inclusion conditions. If a triangle has two equal sides, it belongs to the set of isosceles triangles, else it doesn't.
In language, category membership is defined by the degree of similarity to a prototypical item, and it is gradual, unlike set membership. The prototypical "bread" in American English is a tin-baked spongy loaf leavened with yeast. A Central European style sourdough boule loaf is still quite universally recognized as "bread" even though it is not the first one that comes to mind to an American. Items like challah or a banana bread are farther apart and some people under some circumstances will say "yes" when asked if they are bread, while other people in other contexts will say "no". But if you ask an American baker if a Liege waffle is bread, I expect the answer to be a resounding "no". At the same time, if you ask American bakers what makes a bread bread, the first criterion that comes to mind will probably be "it is a baked good made with yeast" (and a Liege waffle is made with yeast). So, while the criteria do exist, you cannot apply them the same way as you can apply them to sets. ("If it is baked and made with yeast, it is bread").
So of course you can ask for a definition of a category, but have to be aware that not all items which fit the definition are actual members of the category, and that items which don't fit the definition can act as a member of the category (permanently or under special circumstances).
This explains your confusion. You assumed that "if it is food, and comes from the sea, it must be seafood". Unlike a geometrical figure ("if it is a triangle, and has a right angle, it must be a right triangle") seafood does not care for a boolean match of necessary and sufficient conditions. You have to look at the traditional usage and the prototype items of the category.
This is what your brain did naturally the first time, before you started second-guessing your wisdom. Edible seaweed is certainly not considered seafood by native English speakers, at least within the US and British cultures. I don't know the exact prototype item for the category, but my guess would be that it is some subset (mathematically defined this time) of mussels, scallops, shrimps and fish.
If you are interesting in background reading on membership in linguistic categories, I can highly recommend George Lakoff.
Seafood generally applies to crustaceans, fish, cephalopods, and ocean mammals; basically sea animals. Here's a definition from Merriam-Webster that says that: edible marine fish and shellfish. Seaweed would be considered food from the sea, but not necessarily seafood. Seaweeds are plants and algae, so not animals. Marking it as seafood would likely confuse English speakers.
I think this belongs in a different category, given the "classical" definition of seafood belonging to fish, shellfish, etc. May I propose "seagan" or "seagetarian"?
Now, if this is purely a "practical" definition we're seeking here that will get you past customs, go with "vegetable" or "plant material". Warning, though, it might not make it through some Customs inspections due to limitations on what can be transported across borders, regardless of whether you pick a "P.C." name or not.