Intuition behind proof systems
Think of $\Sigma^{*}$ encoding some sort of objects, and $Q$ as the set of all objects satisfying some property. Think of $P$ as a function which accepts (the encoding of) a pair $(x, p)$ where $x$ is an object and $p$ is alleged "evidence" of $x \in Q$. The function $P$ is a "proof checker": it verifies that $p$ actually represents valid evidence that $x \in Q$. If so, it returns $x$, otherwise it returns a default element of $Q$.

As an example, suppose $\Sigma^{*}$ encodes graphs and let $Q$ be the set (of encodings) of Hamiltonian graphs. A possible $P$ is this: decode input as $(G, \ell)$ where $G$ is a graph and $\ell$ is a list of vertices of $G$; verify that $\ell$ is a Hamiltonian cycle in $G$; if so then return $G$ otherwise return the graph on one point.

You considered the case of planar graphs. To get a suitable $P$ we need a notion of poly-time checkable evidence of planarity.

In general the input to $P$ need not encode a pair $(x, p)$. The important thing is that $P$ can extract two pieces of information from its input: the object in question and the alleged evidence that the object belongs to $Q$. For example, let us take as $Q$ the set of all sentences provable in some first-order theory. Now $P$ decodes its input as a formal proof. If the encoding is invalid, it returns $\top$. If the encoding represents a valid proof, it returns the statement that was proved by the proof (which is likely to be the root of proof tree, or the last formula in a sequence of statements, depending on how you formalize proofs).