(This was supposed to be a comment, but I do not have enough reputation right now, so will extend my answer and try to both answer your question and make side comments).

I personally think the question is somewhat ambiguous since I cannot really see how the difference between Platonist approach and formalist approach could be visible in a textbook.

I do not think it would really matter which philosophical view you are siding with, because when it comes to doing mathematics, you have to act like a "formalist" anyway, meaning that you will formalize your argument in some fixed formal system and work there disregarding any notion that cannot be formalized inside.

One of the best formal systems you can formalize (pretty much) everything is ZFC. This means that the "setting" you are looking for when you said

I find it that I understand mathematical theories best when the setting in which the theory "lives" in is clearly outlined so that working in that theory is just formal manipulations of symbols - of course I can attache meaning and intuition to these manipulations...

can easily be taken to be ZFC.

My impression from how you asked your question is that you are looking for a textbook that keeps the distinction between formalized concepts and the intuitive ones we have in mind. For example, we have an intuition for what "truth" means in a structure and we can formalize the corresponding satisfaction relation (say, in ZFC). When you or any logic book prove a theorem involving it (let's take the soundness theorem example you gave), then what you really prove is (formal) provability implies (formal) satisfaction.

The only difference between a Platonist and a formalist stating the same theorem is that the former would see no reason to keep the part in parentheses (for they believe the natural numbers and our formalization of it are the one). You apparently would, as you seem to be not believing in the existence of the natural numbers.

Going back to main question, any textbook that was suggested on MO should be precise enough for you to formalize all model theoretic notions so that you can "keep a formalist point of view" (whatever that means for you).

Specifically, for set theory, I can say that Kunen's book (Set Theory: An Introduction to Independence Proofs) and Jech's book (Set Theory) both make clarifications about formalizations and metatheory time to time, which is sufficient to eliminate confusions.

(This was supposed to be a comment, but I do not have enough reputation right now, so will extend my answer and try to both answer your question and make side comments).

I personally think the question is somewhat ambiguous since I cannot really see how the difference between Platonist approach and formalist approach could be visible in a textbook (except, details of formalizations and keeping proofs and arguments closer to how they would "formally" look).

I do not think it would really matter which philosophical view you are siding with, because when it comes to doing mathematics, you have to act like a "formalist" anyway, meaning that you will formalize your argument in some fixed formal system and work there disregarding any notion that cannot be formalized inside.

One of the best formal systems you can formalize (pretty much) everything is ZFC. This means that the "setting" you are looking for when you said

I find it that I understand mathematical theories best when the setting in which the theory "lives" in is clearly outlined so that working in that theory is just formal manipulations of symbols - of course I can attache meaning and intuition to these manipulations...

can easily be taken to be ZFC.

My impression from how you asked your question is that you are looking for a textbook that keeps the distinction between formalized concepts and the intuitive ones we have in mind. For example, we have an intuition for what "truth" means in a structure and we can formalize the corresponding satisfaction relation (say, in ZFC). When you or any logic book prove a theorem involving it (let's take the soundness theorem example you gave), then what you really prove is (formal) provability implies (formal) satisfaction.

The only difference between a Platonist and a formalist stating the same theorem is that the former would see no reason to keep the part in parentheses (for they believe the natural numbers and our formalization of it are the one). You apparently would, as you seem to be not believing in the existence of the natural numbers.

Going back to main question, any textbook that was suggested on MO should be precise enough for you to formalize all model theoretic notions so that you can "keep a formalist point of view" (whatever that means for you).

Specifically, for set theory, I can say that Kunen's book (Set Theory: An Introduction to Independence Proofs) and Jech's book (Set Theory) both make clarifications about formalizations and metatheory time to time, which is sufficient to eliminate confusions.