In general, the term "use" refers to the act of employing something for a specific purpose or in a specific context. When we use words or formulas, we are employing them to convey a particular meaning or to perform a specific function.
In the case of the formula "x+1=2", it is being used to express the fact that, for a particular value of x (in this case, 1), the formula is true. This is an example of using the formula, as it is being employed to convey a specific meaning.
As for free variables, they can be used in a similar way to other variables. For example, if we have a formula with a free variable x, we can use that formula to represent a relationship between x and other variables in the formula. For example, the formula "x+1=2" can be used to represent the relationship between x and the other variables in the formula (in this case, 1 and 2).
In terms of the syntax and semantics of variables, the syntax refers to the way in which the variables are written or arranged, while the semantics refers to the meaning of the variables. When using variables, both the syntax and the semantics are important. For example, in the formula "x+1=2", the syntax of the formula (i.e., the way in which the variables and operators are arranged) conveys the mathematical operation being performed, while the semantics of the variables (i.e., the meaning of the individual symbols) determines the specific values being used in the formula.
As for the question of intent, whether something is considered "use" or "mention" can depend on the context and the intention of the speaker or writer. In the case of the statement "Some would say 'two is a prime number'", it could be considered either use or mention depending on the context and the speaker's intention. If the speaker is simply mentioning the statement without expressing their own opinion on its truth or falsehood, it could be considered a mention. However, if the speaker is using the statement to make a point or to express their own opinion, it could be considered a use. Ultimately, the determination of whether something is "use" or "mention" depends on the context and the speaker's intent.