Let $M^n$ a smooth manifold and $X \in \mathfrak{X}(M)$ a vector field on $M$. I'm trying to transfer the Tubular Flow Theorem from $\mathbb{R}^n$ to $M$. For this, let $p \in M$ a non-singular point of $X$ and $(U,x)$ a chart over $p$ in $M$. The question is:
How to transfer "correctly" $X|_U=\sum a_i\dfrac{\partial}{\partial x_i}$ from $U$ to $x(U)\subset \mathbb{R}^n$?
My idea is define, for $u \in x(U)$, a vector field $Z(u)=(a_1(x^{-1}(u)),\dots,a_n(x^{-1}(u)))$. I think this is OK, in the textbook I'm using, the author uses notation $x_*X$ for a "vector field induced by $X$ on $x(U)$" but does'nt defines what precisely it is...
Someone have a definition for this? My definition of $Z$ agrees with $x_*X$?
Technically, what you call $X\vert_U$ seems already to be the vector-field on $x(U)$.
$x_*X$ is the pushforward of $X$ via $x$, i.e. is the vector-field such that, given $f:x(U)\to\mathbb R$, you have $$x_*X (f)=X(f\circ x)\;.$$
When you write $$\sum a_i\frac{\partial}{\partial x_i}$$ you are writing a vector-field on $\mathbb R^n$ with respect to the coordinates $x_1,\ldots, x_n$, because that derivation/vector-field/partial derivative with respect to $x_i$ makes sense only in $\mathbb R^n$.
The only meaning this writing has on a manifold is with respect to local coordinates, that is, in a chart, where you define vector-fields $\partial_1,\ldots,\partial_n$ in $\mathfrak{X}(U)$ such that $$x_*\partial_j=\frac{\partial}{\partial x_j}\;.$$
Now, in the same way, one usually writes coefficients $a_1,\ldots, a_n$ as functions of the local coordinates, so you will already have $a_j=a_j(x_1,\ldots, x_n)$, because the frame $\partial_1,\ldots, \partial_n$ exists only on $U$, so the writing $$X=\sum a_j\partial_j$$ makes sense only on $U$. Therefore, it is harmless to suppose that $a_j$ is a function of $x_1,\ldots, x_n$.
With these assumptions (which are the only reasonable ones under which your expression makes sense), the coordinates of $x_*X$ in $\mathbb R^n$ are exactly the functions $a_1,\ldots, a_n$, viewed as functions of $x_1,\ldots, x_n$.
If you instead for some reason want to remember that $a_1,\ldots, a_n$ are functions on $U$, then, you will write, as you say, $a_j\circ x^{-1}$.
Note: I strongly doubt that any book about vector-fields can "forget" to define the pushforward, so I would suggest you to read more carefully the previous pages (or chapters), in particular when the definition of vector-field is given.