Short answer: Your confusion stems from not understanding formal systems and their objectivity. Provability in a formal system is almost completely objective, more objective that anything else you can ever hope for. So logical consequence within a formal system is equally objective.
A mathematically proven statement would be absolutely correct if all the axioms and inference rules used in the proof are first accepted as absolutely correct. That is the whole purpose of creating formal systems, so that anyone who accepts all the axioms and inference rules will have no choice but to accept all the statements proven within the formal system.
In the above quote from this post, "accept all the axioms and inference rules" means "accept all the axioms and accept all statements produced by applying any of the inference rules to any accepted statement".
Note that this has almost nothing to do with semantics (i.e. meaning of words). It does not matter what you mean by "accept". It does not matter if you interpret statements differently from me. It only matters that exactly the above is satisfied, namely the statements that you accept include the axioms and are closed under applying inference rules. I say "almost nothing" because we do need to share a very tiny bit of understanding, but that is solely understanding of what each inference rule means, and nothing more. For example, the ∧-intro rule in all natural deduction systems means that if you accept A and accept B then you accept (A∧B). If you want to be 100% precise, in most formal systems it means that if you accept the string A and the string B then you accept the string "("+A+")∧("+B+")", where brackets are used to ensure correct parsing. Some systems may precedence rules, some may not. None of that matters; all that matters is that you understand exactly what each inference rule means, which is why it is so crucial that every formal system can be captured completely by purely syntactic rules, because that guarantees that we can achieve such a shared understanding.
Even if whenever you say "I accept X" you actually mean "I doubt X", it makes no difference at all to the outcome. Suppose you say "I accept the axiom A" and "I accept the axiom B" and "I accept the ∧-intro rule". Then I can by applying the ∧-intro rule guarantee that you have to admit "I accept the statement (A∧B)". I do not need to even know what you mean by "accept", to be able to objectively demonstrate that you must accept every statement that is proven within a formal system whose axioms and rules you accept, as long as I provide the proof (i.e. the sequence of steps each of which states which rule to apply to which statements).
In the past, one may be forgiven for questioning whether even syntactic rules are sufficient to achieve shared understanding of formal systems. But ever since we have had personal computers, that is no longer tenable, because every formal system can be literally implemented by a computer program in a fixed programming language. This even generalizes the notion of syntactic rules in formal systems. An inference rule can be defined as a program that when run on input statements will produce an output statement! Accepting a set A of axioms and a set R of inference rules can then be simply defined as accepting the closure C of A under R. We can very well call C the logical consequences of A under R, and there is essentially no subjectivity concerning whether a statement X is a member of C once I write down explicitly a sequence of steps each specifying which rule in R to run on which statements, because you can completely mechanically check each step by running the specified programs!
The only possible subjectivity in proofs within a formal system is if you doubt your own ability to apply the stated inference rules (i.e. run the given programs) correctly. At that point, it is far more reasonable to doubt that your browser has correctly fetched this webpage for you. (Maybe your browser program had a strange bug/feature and added this sentence which user21820 did not write!)