In model theory the notion of categoricity captures the concept I think you are looking for. In particular Lefschetz principle is equivalent to the fact that the theory of algebraically closed fields is categorical. By this we mean that any two algebraically closed fields of the same cardinality are automatically isomorphic and in fact, any first order statements about such a field only depends on this cardinality.

http://en.wikipedia.org/wiki/Morley's_categoricity_theorem

There are model theoretic statements about complete discrete valuation rings which only depend the residue field. I'm not sure if this is what you are interested it.

The statement as you stated it is false: If k =FFbar_p. Every local rings takes the form W(k)[[X_1, \ldots, X_n]]/I for some ideal I by the Cohen-Structure theorem. Here W(k) is the ring of p-typical witt vectors of k. Then the first order sentence 1 + ... + 1 =0 (p-times) is true for analytic rings while it is not true for W(k).

Suppose you want to refine the statement, by just when k is the complex numbers.

If k=CC, and you allow me to make statements about convergence (using absolute values on CC in my language) I don't think it would be hard to come up with a first order statement which would be true for elements of analytic rings but not true for CC[[T]].

You can refine further and this leads to restricting what kind of statements are allowed (essentially restricting this symbols we are allowed to use). This can lead to thinking about things like the model theory of valued fields which is quite a large body of mathematics. I'm not an expert in this so I can just point you in this direction.

In model theory the notion of categoricity captures the concept I think you are looking for. In particular Lefschetz principle is equivalent to the fact that the theory of algebraically closed fields is categorical. By this we mean that any two algebraically closed fields of the same cardinality (and characteristic, provided it is at least the continuum) are automatically isomorphic and in fact, any first order statements about such a field only depends on this cardinality. (The countable algebraically closed fields are isomorphic to algebraic closures of QQ(t_1,...,t_n) and transcendence degree distinguishes these)

http://en.wikipedia.org/wiki/Morley's_categoricity_theorem

There are model theoretic statements about complete discrete valuation rings which only depend the residue field. I'm not sure if this is what you are interested it.

The statement as you stated it is false: If k =FFbar_p. Every local rings takes the form W(k)[[X_1, \ldots, X_n]]/I for some ideal I by the Cohen-Structure theorem. Here W(k) is the ring of p-typical witt vectors of k. Then the first order sentence 1 + ... + 1 =0 (p-times) is true for analytic rings while it is not true for W(k).

Suppose you want to refine the statement, by just when k is the complex numbers.

If k=CC, and you allow me to make statements about convergence (using absolute values on CC in my language) I don't think it would be hard to come up with a first order statement which would be true for elements of analytic rings but not true for CC[[T]].

You can refine further and this leads to restricting what kind of statements are allowed (essentially restricting this symbols we are allowed to use). This can lead to thinking about things like the model theory of valued fields which is quite a large body of mathematics. I'm not an expert in this so I can just point you in this direction.