My interpretation of this is that there really is no such thing as the "storm count". It's a convenience to the players while playing the game. Compared to life totals, Storm is an activated ability and not a critical component of the game like life.
There is such a thing as "Storm Count." It's not an official term, no, but it simply refers to the number of spells that have been cast during the current turn. Sure someone could keep track of "Number of spells cast this turn" and keep referring to it as such, but storm is (as far as I can recall) the only thing it matters for, so calling it "storm count" is easier for most people. (Also, storm is a triggered ability, not activated.)
A player is not required to provide their opponent with a running total of how many spells have been played during the turn. Many do so because it's convenient, and typically doesn't provide your opponent any kind of advantage. If you think it will for some reason (certainly is possible), you can always keep the count to yourself.
Making sure you and your opponent are clear and in agreement on something that you know will be relevant later (even if your opponent doesn't know this) also has the advantage that it's less likely there will be a dispute on it, and that if there is a dispute and a judge gets involved, less likely the judge will accept your opponent's version as correct. Think about it, if you cast 10 spells, graveyards get shuffled around, and such, and then when a judge gets called over, you explain, "I cast Ancestral, then 2 other spells, but I don't remember what, then..." your lack of remembering details may work against you.