"A" before "soup"

[teechar]

Type of is implicit in all these sentences,

No. "food" is used in its generic sense in those sentences. They're like definitions.
What is fresh food?
Fresh food is: food which has not been preserved.

[Alexey86]

No. "food" is used in its generic sense in those sentences. They're like definitions.
What is fresh food?
Fresh food is: food which has not been preserved.

So, fresh food is not a (type of) food which has not been preserved, right?

[teechar]

Correct. In that sentence, it's not a type of food. It is any food (which has not been preserved).

[Alexey86]

Correct. In that sentence, it's not a type of food. It is any food (which has not been preserved).

Just to clarify, if I wrote Fresh food is a food which has not been preserved, would it be incorrect, or would a be read as a short form for a type of?

There are also the following sentences in the article:

A negative-calorie food is food that supposedly requires more food energy to be digested than the food provides.
A functional food is a food given an additional function by adding new ingredients or more of existing ingredients.

These are also definitions, but for some reason they take a. And what is confusing is that the latter takes a after is, while the former doesn't.

[GoesStation]

Just to clarify, if I wrote Fresh food is a food which has not been preserved, would it be incorrect, or would a be read as a short form for a type of?

It would be clumsy at best.

A negative-calorie food is food that supposedly requires more food energy to be digested than the food provides.
A functional food is a food given an additional function by adding new ingredients or more of existing ingredients.

These are also definitions, but for some reason they take a. And what is confusing is that the latter takes a after is, while the former doesn't.

The first one is incorrect.

[Alexey86]

It would be clumsy at best.

A fresh food is a food which has not been preserved.

Is this variant appropriate?

[GoesStation]

A fresh food is a food which has not been preserved.

Is this variant appropriate?

Yes.

[Alexey86]

Yes.

Ramen is a spicy soup.
Fresh food is a food which has not been preserved.

Why is a perfectly correct in the former meaning a type of and inappropriate in the latter? Both statements are general.

[Rachel Adams]

Where are you seeing a difference? All the responses so far have indicated that "a" is required in your sentences.

You used "see" in the progressive. Can we non-native speakers use it too?

[GoesStation]

You used "see" in the progressive. Can we non-native speakers use it too?

I don't see why not, provided you use it idiomatically. I don't think there are many cases where it's natural.