Modus ponens is stated $$((P \implies Q) \land P) \implies Q$$ But isn't $(P \implies Q) = (\lnot P \lor Q)$?
Then we get $((P \implies Q) \land P) \implies Q$ $= (\lnot P \lor Q) \land P$ $= (\lnot P \land P) \lor (P \land Q)$ $= \text{False} \lor (P \land Q)$ $= P \land Q$ So $P\land Q$, not just $Q$. Why do we say that modus ponens then just implies $Q$?
Yes, indeed, $P$ and $Q$ are both implied by $P$ and $P\to Q$. In fact, $P$ and $Q$ and $P\to Q$ are all implied by $P$ and $P\to Q$.
However, we're most interested in learning new things. $Q$ is new thing we learn from knowing $P$ and $P\to Q$. It is not the only thing, but it is the new thing.
It is useful to know this, so we give the rule of inference a special name: "modus ponens".