It is important to stick to mathematical logic here as opposed to the definitions from philosophy. The first sentence of the wikipedia
In mathematical logic, a logical system has the soundness property if
and only if every formula that can be proved in the system is
logically valid with respect to the semantics of the system.
while not fully precise, is the sense in which 'sound' is generally used in mathematical logic. This corresponds to a system having valid axioms and validity-preserving rules of inference, so you can see how it corresponds loosely to the philosophical notion of true premises and correct (i.e. truth-preserving) argumentation.
As for validity, the wikipedia sentence
A formula of a formal language is a valid formula if and only if it is
true under every possible interpretation of the language. In
propositional logic, they are tautologies.
uses 'valid' in the way it is used in mathematical logic. Here the connection to the philosophical notion of a valid argument that you cite is a little less direct. Let $A$ and $B$ be your premises and $C$ your conclusion. And say $$ (A\land B) \to C $$ is valid in the mathematical sense. That means that $A\land B \to C$ is true in any interpretation, which means that it is impossible for $A$ and $B$ to be true and for $C$ to be false. So the two notions of validity connected, but even more loosely than was the case for soundness.
Note that the validity of $(A\land B)\to C$ is the same thing as $C$ being a semantic consequence of $A$ and $B,$ i.e. $$ A,B\models C$$ which has more of the flavor of "a valid argument from $A$ and $B$ to $C$," though strictly speaking, there's no 'argument' here. (However, the syntactic variation $A,B\vdash C,$ which is equivalent in the presence of a completeness theorem, means there's a proof of $C$ from assumptions $A$ and $B$... that's more of an 'argument'.)
I've tried to be conciliatory to the philosophical definitions here, but in math it's important to get comfortable with the fact that precise definitions vary from treatment to treatment. (Heck, the other day I learned that not all books have the same definition of 'compact set' in a general topological space, which was a definition that I thought was more or less sacrosanct.) It should also go without saying that the definitions don't need to conform to the colloquial meaning of the words, though it's nice when they're at least not wildly misleading.
In a field with as many moving parts in mathematical logic we will often need to adjust and adapt our terminology as we develop new ideas and try to apply mathematical logic to more exotic situations. Even in the beginning of my answer where I gave some 'general purpose' definitions and remarks, I had some particular contexts in mind, and we will need to make these things fully precise in any detailed treatment. For instance, when we move to the specific context of classical propositional logic, "valid" becomes synonymous with "tautology" and we make precise the ideas of "interpretation" and "truth" in a given interpretation. When we flesh things out in a given context (especially when working in more exotic situations than, say, classical propositional or predicate logic) everything I said is subject to revision.