A clarification of this model theory question regarding when two theories are the same I was told in an answer to a previous question that for all non-zero real numbers $r$ and $r'$, the theory of the structures $(\mathbb{R};+,r)$ and $(\mathbb{R};+,r')$ are the same. But how can this be? For example, the theory of $(\mathbb{R};+,2)$ is different from the theory of $(\mathbb{R};+,3)$, because, for example, the first contains the statement $2=2$ while the second one doesn't, because it doesn't have a $2$ symbol in the first place. So, are those theories really the same, or merely equivalent, and if so, what is the definition of two theories being equivalent?
I would like a clarification of this issue.

Neither theory contains the statement $2 = 2$ because $2 = 2$ is not a statement in the vocabulary.
You have discussed structures, but structures must be a structure over some language. The language here is a language consisting of one binary operator (which we'll call $+$) and one constant symbol (which we'll call $c$).
The informal statement $2 = 2$ corresponds to the formal proposition $c = c$, which is clearly satisfied by both structures.
Because there is an isomorphism between the structures $(\mathbb{R}; +, r)$ and $(\mathbb{R}; +, r')$, they have the same theory.
