Let's say we have an algebraic number field $K$ and a point $Q\in X_1(N)$ that is not a cusp. Now $Q$ can be represented as $Q=[(E,P)]$, where $E$ is an elliptic curve and $P$ is a pont on $E$ of order $N$. I know that there exists such pair $(E,P)$ such that $E$ is defined over $K$, but I'm not sure if a pair $(E,P)$ where $P$ is defined over $K$ always exists.
EDIT: example for clarity.
For example, if we have $$E: y^2=x^3+2$$ and $P=(0, \sqrt{2}).$ Then we have that $Q=[(E,P)]\in X_1(3)(\mathbb{Q})$ since every $\sigma \in G_\mathbb{Q}$ fixes the point $Q$. Namely, if $\sigma\in G_\mathbb{Q}$ then $E^\sigma =E$ and $P^\sigma =\pm(0,\sqrt{2})$. In any case we have $[(E,P)]=[(E^\sigma, P^\sigma)]$. Therefore $Q=[(E,P)]\in X_1(3)(\mathbb{Q})$.
However, it can be shown that $[(E,P)]=[(E', P')]$ (i.e., $E$ and $E'$ are isomorphic and that isomorphisam sends $P$ to $P'$), where $$E': y^2=x^3+1$$ and $P'=(0,1)$. We see that now $P$ is defined over $\mathbb{Q}$ as well. Can we always find such $P$?
If $K$ is fixed, it is not garunteed to exist except for some finite collection of $N$ depending on the number field $K$. When $K = \mathbb{Q}$ this is due to Mazur and when $K$ is a general number field this is extended by Merel.
The simplest case is when $N = 11$ and $K = \mathbb{Q}$, then $X_1(11)$ is the elliptic curve with Cremona label $11a1$. This elliptic curve has rank $0$ and the torsion subgroup is a cyclic group of order $5$. Each of these $5$ rational points is a cusp - so there exists no non-cuspidal rational point on $X_1(11)$, hence there exists no pair $(E/\mathbb{Q}, P)$ where $P \in E(\mathbb{Q})$ has order $11$.
To clarify, if $K/\mathbb{Q}$ is a number field the action of the Galois group $G_K$ on $X_1(N)$ is natural in the following way: If $Q \in X_1(N)(\bar{\mathbb{Q}}) \setminus \{cusps\}$ then $\sigma \in G_{K}$ acts on the pair $(E, P)$ representing $P$ by taking $(E, P)^\sigma = (E^\sigma, P^\sigma)$. Hence if $Q$ is stable under $G_K$ (i.e., is $K$-rational) if and only if both $E$ and $P$ are stable under $G_K$ - that is both $E$ and $P$ are defined over $K$.
Thus in your example, the point $(E, P)$ is not defined over $\mathbb{Q}$, since there are elements of galois which send $P \mapsto -P$ - the specific point does matter. Your argument does show that this example supports a point on $X_0(3)$ since $\{O, P, -P\}$ is a galois stable cyclic subgroup of $E[3]$.
Another way to view this is if you just define $Y_1(N)/\mathbb{Q}$ to be a scheme which represents the moduli functor $S \mapsto \{(E/S, P) : P \in E(S) \text{ has order $N$}\}$.