About change of coordinates of vector fields in smooth manifold theory

Background

Suppose $$M$$ is a smooth $$n$$-manifold and $$(U,\varphi=(x^i))$$ a smooth chart on $$M$$. For each point $$p\in U$$ we know that $$(\frac{\partial}{\partial x^i}|_p)_{i=1}^n$$ is a basis for the tangent space $$T_pM$$. If $$(\tilde U,\tilde\varphi=(\tilde x^i))$$ is another smooth chart on $$M$$ with $$U \cap \tilde U\ne \emptyset$$, then for each point $$p\in U \cap \tilde U$$ we have the following formula for the basis - change:

$$(\frac{\partial}{\partial x^1}|_p,\dots,\frac{\partial}{\partial x^n}|_p)=(\frac{\partial}{\partial \tilde x^1}|_p,\dots,\frac{\partial}{\partial \tilde x^n}|_p) \cdot \operatorname{Jac}(\tilde\varphi\circ\varphi^{-1})_{\varphi(p)} \quad$$

where $$\operatorname{Jac}(\tilde\varphi\circ\varphi^{-1})_{\varphi(p)}$$ is the Jacobian matrix of $$(\tilde\varphi\circ\varphi^{-1})$$ in $$\varphi(p)$$.

Fact

Suppose we have the following vector field on $$\mathbb{R}^2$$: $$X=2x \frac{\partial}{\partial x}$$. Let $$f:\mathbb{R}^2 \to \mathbb{R}, (x,y)\mapsto x^2$$.

We want to compute the coordinate expression for $$X$$ in polar coordinates (on some open subset on which they are defined) and show that it is not equal to $$\frac{\partial f}{\partial r}\frac{\partial }{\partial r}+ \frac{\partial f}{\partial \theta}\frac{\partial }{\partial \theta}$$.

My understanding and execution

Let $$M=\{(x,y)\in \mathbb{R}^2 : x>0\}$$, and

$$N=(0,+\infty)\times (-\frac{\pi}{2},\frac{\pi}{2})$$.

Let $$F=M\to N, (x,y)\mapsto (\sqrt {x^2+y^2},\arctan\frac{y}{x})$$.

Then $$F$$ is a diffeomorphism with inverse $$F^{-1}:N \to M, (r,\theta)\mapsto (r\cos\theta, r\sin\theta)$$.

Issue 1: Polar coordinates are, by definition, the component functions of the map $$F$$, right? So we have $$F=(r, \theta)$$.

Great Issue 2: I have an ambiguity on how to interpret $$\frac{\partial}{\partial r}$$ and $$\frac{\partial}{\partial \theta}$$.

(a) Should I interpret $$(r,\theta)$$ also as standard coordinates on $$N$$ and thus $$\frac{\partial}{\partial r}$$ and $$\frac{\partial}{\partial \theta}$$ are simply the ordinary partial derivatives on $$N$$? (Just like $$\frac{\partial}{\partial x}$$ and $$\frac{\partial}{\partial y}$$ on $$M$$?)

(b) I know that if $$M$$ is a smooth $$n$$-manifold and $$(U,\varphi=(x^i))$$ is a smooth chart on $$M$$, then for each point $$p\in U$$ we have that $$\frac{\partial}{\partial x^i}|_p=d\varphi^{-1}_{\phi(p)}(\partial_i|_{\phi(p)})$$ where $$\partial_i$$ is the standard $$i$$-th partial derivative on $$\mathbb{R}^n$$. So thinking of my $$F$$ as a smooth chart, I should interpret $$\frac{\partial}{\partial r}|_{(x,y)}=dF^{-1}_{F(x,y)}(\frac{\partial}{\partial x}|_{F(x,y)})$$ for each $$(x,y)\in M$$.

Which one between (a) and (b) is the correct interpretation?

Issue 3 The vector field I'm looking for, say $$Y$$, should be the pushforward of $$X$$ by $$F$$?

Issue 4 Applying formula  I have

$$(\frac{\partial}{\partial x}|_{(x,y)},\frac{\partial}{\partial y}|_{(x,y)})=(\frac{\partial}{\partial r}|_{(x,y)},\frac{\partial}{\partial \theta}|_{(x,y)}) \cdot \operatorname{Jac}(F)_{(x,y)}$$ which i can rewrite

$$(\frac{\partial}{\partial x}|_{(r\cos\theta,r\sin\theta)},\frac{\partial}{\partial y}|_{(r\cos\theta,r\sin\theta)})=(\frac{\partial}{\partial r}|_{(r\cos\theta,r\sin\theta)},\frac{\partial}{\partial \theta}|_{(r\cos\theta,r\sin\theta)}) \cdot \operatorname{Jac}(F)_{(r\cos\theta,r\sin\theta)},$$ from which I have

$$\frac{\partial}{\partial x}|_{(r\cos\theta,r\sin\theta)}=\cos\theta \frac{\partial}{\partial r}|_{(r\cos\theta,r\sin\theta)}-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}|_{(r\cos\theta,r\sin\theta)};$$ finally we have $$X=2x\frac{\partial}{\partial x}=2r\cos\theta\frac{\partial}{\partial x}|_{(r\cos\theta,r\sin\theta)}=2r\cos^2\theta \frac{\partial}{\partial r}|_{(r\cos\theta,r\sin\theta)}-2\sin\theta\cos\theta\frac{\partial}{\partial \theta}|_{(r\cos\theta,r\sin\theta)}.$$ I think there is something wrong with this and whit using formula  in this context. Instead, using the formula for computing the pushforward of $$X$$ by $$F$$, I got $$(F_*X)_{(r\,\theta)}=2r\cos^2\theta \frac{\partial}{\partial r}|_{(r,\theta)}-2\sin\theta\cos\theta\frac{\partial}{\partial \theta}|_{(r,\theta)}$$ which sounds correct.

So is there some "abuse of notation" which is confusing me? Am I right in saying that strictly speaking I cannot use formula ?

Please excuse me for the very long post, I have done the possible to make it clear.