I'm having a little trouble formalizing the idea of using "ambient standard coordinates" on level sets of $\mathbb{R}^n$. For example, we typically write $S^1=\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\}$, which can be seen as the (regular) level set $f^{-1}(1)$ associated to the smooth function $f(x,y)=x^2+y^2$. From what I understand, we can view these $x,y$ as the coordinate functions that come from the standard coordinate chart $\varphi$ on $\mathbb{R}^2$.
Suppose I wanted to define a vector field $X$ on a (connected) open subset of $S^1$. In this case, I'd be working with the usual angle coordinate $\theta:U\longrightarrow \mathbb{R}$, and so this vector field can be written on $U$ as $X_p=h(p)\frac{d}{d\theta}$. Essentially, what formally/rigorously allows us to also express this vector field as (say) $$X_p=y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}$$ using the standard coordinates on $\mathbb{R}^2$? The angle coordinate $\theta:U\longrightarrow \mathbb{R}^1$ has one coordinate function, so I'm not sure how we can apply the change of coordinates formula in Lee's Introduction to Smooth Manifolds (see pg. 63), since $\varphi=(x,y):\mathbb{R}^2\longrightarrow\mathbb{R}^2$ has two coordinate functions. I suppose I'm intuitively comfortable working with these standard coordinates (say, in the sense of multivariable calculus), but I think it would be especially helpful for my understanding how this intuition is formalized.
A small footnote: when I expressed $X_p$ in terms of standard coordinates, I was using the fact that $T_pS^1$ consists of vectors which are orthogonal to the point $p=(x,y)$. I'm able to see why I can do this by examining $\ker df_p$.
With respect to a local chart $(U,\theta)$ of $S^1$ and the global chart $\mathrm{id}=(x,y)$ of $\mathbb R^2$, the standard inclusion $i:S^1\to \mathbb R^2$ looks like $$(x,y)(i(p))=(\cos(\theta(p)),\sin(\theta(p))).$$ This embedding induces a tangent map $Ti:TS^1\to T\mathbb R^2$, whose restriction to $TU\subseteq TS^1$ looks like $$Ti\left(a\frac{\partial}{\partial \theta}\bigg|_{p}\right)= a\left( -\sin\theta(p)\frac{\partial}{\partial x}\bigg|_{i(p)} + \cos\theta(p)\frac{\partial}{\partial y}\bigg|_{i(p)} \right).$$ This map is injective at every point $p$ of $u$, and it is the identification of $T_pS^1$ with a subspace of $T_{i(p)}\mathbb R^2$ that jd27 was talking about.