The similarity of American and European legal terms are deceptive. In US legal documents each word is carefully chosen and has a specific meaning. For instance, verbs that are often used include ‘certify’ or ‘attest’. In European minds these are just synonyms for ‘declare’, ‘state’, or something to that extent. One’s English isn’t necessarily that impeccable to be bothered with minute details, and a signature is scribbled under it without further thought.
And indeed in the European legal world being creative with words does not necessarily have far-reaching consequences. A contract is not necessarily drafted by an attorney but can be written by the sales department or other non-legal division. After all the relevant law, a statute, provides the exact meaning. Culturally, European writing style rules don’t allow using the same word twice in the same paragraph. Sometimes words are avoided purely because they sound too complicated, foreign, ugly, don’t fit in the corporate style or have a cultural meaning that is different from what the contract is intending to convey. Oddly - considering that a contract is actually a legal document - a word that sounds ‘too legalese’ is often replaced by a term that is better understood by a layman. This ensures that individuals in e.g. the sales or manufacturing or accounting department can actually work with the document. But if this happens in the US the attorneys have a field day, and if it ever comes to litigation it ensures them months of extra work.
In America there is no central statute that provides the proper meaning or rule. That is one of the reasons that attorneys all use the same terms and don’t randomly change words with synonyms. If the language in a document is not aesthetically appealing, soft. It doesn’t matter that a layman can’t read it; in the end it is a document that serves as ‘law. It is not designed for non-attorneys to work with; there is no room for legal ambiguities. If it is not clear to a non-lawyer one can always ask an attorney for clarification. In that context, if distinct verbs such as ‘affirming’, ‘certifying, ‘attesting’ are used, if the declaration turns out not to be true, chances are that the person making the statement has misrepresented the situation, sometimes even conducted perjury, or will otherwise be subject to charges.