Consider the Lie derivative of the vector field $\bf{Y}$ with respect to the vector field $\bf{X}$ on manifold $M^{n}(x)$ defined as
$$\displaystyle{[\mathcal{L}_{\bf{X}}Y]_{x}:=\lim_{t\rightarrow 0} \frac{[{\bf{Y}}_{\phi_{t}x}-\phi_{t*}{\bf{Y}}_{x}]}{t}}$$
${\bf{Y}}_{\phi_{t}x}$ is the tangent vector of the vector field $\bf{Y}$ at the point $\phi_{t}x$, where the point $\phi_{t}$ is obtained by starting at point $x$ at time $0$ and traversing along the orbit of $x$ to time $t$.
But I don't understand how to interpret $\phi_{t*}{\bf{Y}}_{x}$. Given the map $\phi_{t}$ which maps points $x$ in $M^{n}$ to points $\phi_{t}(x)$ in $M^{n}$ along the orbit of $x$ parameterised by time $t$, we can define the differential $\phi_{t*}$ that maps the tangent vector of the vector field $Y$ at $x$ to some tangent vector at the point $\phi_{t}x$. Now, there is only one tangent vector of the vector field $\bf{Y}$ at the point $\phi_{t}x$, and this tangent vector is the vector ${\bf{Y}}_{\phi_{t}x}$. This seems to suggest that ${\bf{Y}}_{\phi_{t}x}$ and $\phi_{t*}{\bf{Y}}_{x}$.
What am I missing?