At common law, the heart of a contract is consensus 'ad idem' - agreement to the same thing. I can only enforce a contract against you if a reasonable person in my position, at the time of contracting, would have believed that you had accepted the contract (including the same terms that I want to enforce).
Usually, having a written contract that both parties sign means that condition's easily met. But pretty clearly, swapping some small print out just before you sign might undermine that, depending on the situation. See e.g. Hartog v Colin & Shields [1939] 3 All ER 56.
(Obviously, this is irrelevant to the person in the article, as Russia is not a common-law country. And I don't know if it's relevant to American HN readers either: my possibly flawed understanding (as a Brit) is that most consumer contracts in the US are not governed by common law, but by the UCC, of which I know nothing).
In US contract law governed by common law principles there is something called the mirror image rule. An offer must be wholly accepted or else rejected. If any terms are modified that is treated as a rejection and counter-offer.
The UCC, which in the relevent part only covers the sale of goods, abrogates this rule and allows and contract to be formed so long as the material terms are identical. The provisions of the resulting contract are governed by UCC 2-207 and the process of determine them is given the colorful name the "battle of the forms".
That's the beauty of using objective (rather than subjective) intent as the test. You don't need to prove belief. Subjective belief is irrelevant. What you need is objective facts, and a court that pretends it's a reasonable person in the position of e.g. the offeree, knowing everything they knew, and decides what it would have understood by the offer.
The point is: you interpret words as they'd be understood by a reasonable person in the position of whoever the words were spoken to, not as they were understood by the person who spoke them.
E.g. consider the Hartog case. Key sentence: "...the plaintiff could not reasonably have supposed that that offer contained the offerers' real intention." ("Could not reasonably have supposed" is legal shorthand for "A reasonable person in their position would not have supposed"). If they had genuinely thought that the offeror meant it, it wouldn't have mattered.
No test by fire needed.
(NB this is English common law. I've been told in the past (https://news.ycombinator.com/item?id=5402595) that the UCC in the US is a whole lot more subjective than English law, so this is likely wrong in the US).
Do you have in UK these scammers that invite bunch of old people to some sort of meeting where they give them cake, some crappy gift and convince them that some other crappy thing, like a wool blanket is worth few thousands dollars and they sign agreements with them so they can pay for the said blanket in installments?
The rule you described is really nice but is it used in practice to fight such scammy business practices?
I've never heard of it, but I wouldn't be surprised if we did.
But: It's not clear to me that the rule is at all relevant to that scam. As you describe it, it's not that the victims reasonably appeared to be agreeing to do one thing when what they signed said something different. They were persuaded to buy a blanket for x thousand dollars, and did.
The most obvious tool contract law has to undermine that is misrepresentation. If the scammer told a lie to get the victim to sign, then the contract's voidable by the victim. (This is true even if the misrep is innocent. If it's fraudulent, then you get a better measure of damages - and, of course, the scammers might've committed the criminal offence of fraud).
There's also a doctrine of 'undue influence', which if proved can also make the contract voidable.
Finally, even if the contract is valid, since R v Hinks [2000] UKHL 53, it's not impossible that the scammers could be convicted of theft of the money. So, validly transferred per civil law, theft per criminal law! Which is... surprising. (IMHO it's bloody stupid, but that's another debate).
Usually, having a written contract that both parties sign means that condition's easily met. But pretty clearly, swapping some small print out just before you sign might undermine that, depending on the situation. See e.g. Hartog v Colin & Shields [1939] 3 All ER 56.
(Obviously, this is irrelevant to the person in the article, as Russia is not a common-law country. And I don't know if it's relevant to American HN readers either: my possibly flawed understanding (as a Brit) is that most consumer contracts in the US are not governed by common law, but by the UCC, of which I know nothing).