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(Please excuse if this questions sounds dull; we are not professional chemists)

In our office, we discussed today what the exact definition of a salt is and whether an acid itself can be classified as a salt.

Our first problem was that we couldn't get a definitive definition for what exactly a salt is. Wikipedia states

In chemistry, a salt is an ionic compound that can be formed by the
neutralization reaction of an acid and a base.

So, let's take, for example, carbonic acid. It is a compound of the ions $ceH+$ and $ceHCO^3-$. We could say that this compound forms due to the reaction of an acid $ceHCO^3-$ and a base $ceH_3O+$, i.e.,

$ceH_3O+ + ceHCO^3- rightarrow ceH_2CO_3 + ceH_2O$

Thus, it seems that the acid fulfills the definition above. We are pretty sure that acids are not considered salts. So where is the error in our reasoning?

It is a semantics question with an open ended discussion. If you recall the old problem "How many angels can dance on a pinhead? Medieval problem, this issue here is similar.

Basically in chemistry, all the terminology is controlled or endorsed by the International Union of Pure and Applied Chemistry (IUPAC). They have a very open ended definition of a salt: "A chemical compound consisting of an assembly of cations and anions."

If we follow some really old chemistry texts (>1850s), yes an acid was defined as a salt of hydrogen ion, i.e., an acid is a salt whose electropositive element is hydrogen Definition of an acid.

Did this definition by a few chemists become popular? The simple answer is no. Not in a general sense.

The modern interpretation of an acid, again from IUPAC is, "A molecular entity or chemical species capable of donating a hydron (proton) or capable of forming a covalent bond with an electron pair."
If you have heard of ionic liquids, they consist of a large organic cation and an anion, this is a salt as well.

Then you have so-called acid salts, such as sodium hydrogen sulfate e.g., $ceNaHSO4$. Their pH is very acidic and these "salts" will neutralize bases like a typical acid.

Thus one may loosely say, that all acids may be considered as salts but not all salts are acids.

It's great that you are discussing chemistry in your office! You write:

[..] carbonic acid. It is a compound of the ions $ceH+$ and $ceHCO^3-$

This is the error in your reasoning. There is a covalent bond between the hydrogen and the rest of the molecule. When it reacts as an acid, this bond breaks.

I like definition d) of the online MW:

any of various compounds that result from replacement of part or all of the acid hydrogen of an acid by a metal or a group acting like a metal : an ionic crystalline compound

So starting with the acid $ceH2CO3$ (carbonic acid), you would let it react as an acid (in water, perhaps), and replace the lost hydrogen ion by a sodium ion, giving you $ceNaHCO3$, sodium bicarbonate. If you isolate that as a pure solid, you get a typical salt (ionic crystals, dissolve more or less in water).

If you want to get back to your definition, you would add carbonic acid and sodium hydroxide in water. Carbonic acid would act as an acid, sodium hydroxide would act as a base, and depending on the mixing ration (stoichiometry), you would get sodium carbonate or sodium bicarbonate. Removing water would give you the ionic compounds.

Now you can go back to discussing how to popcount large arrays...