I find that using the notation $d\phi^{-\epsilon}$ and so on quite unhelful. Instead, given vector fields $X$ and $Y$ I imagine the vector $Y$ at $x$ as a small arrow painted on a fluid whose velocity field is $X$. The tail of the arrow is at $x$ and its head at $x+\eta Y$, where $\eta$ is a small number. After the short time $\epsilon$ the tail of the $Y$ arrow has been carried by along by the fluid to $x+\epsilon X$ and its head is wherever it has been carried to. One subracts the flow-carried $Y$-arrow from the value of the vector field $Y(x+\epsilon X)$. The latter is represnted by the small arrow whose tail is at $x+\epsilon X$ and whose head is at $(x+\epsilon X)+\eta Y(x+\epsilon X)$. Then you divide by $\epsilon$ and by $\eta$. The result is ${\mathcal L}_XY$

I find that using the notation $d\phi^{-\epsilon}$ and so on quite unhelpful. Instead, given vector fields $X$ and $Y$ I imagine the vector $Y$ at $x$ as a small arrow painted on a fluid whose velocity field is $X$. The tail of the arrow is at $x$ and its head at $x+\eta Y$, where $\eta$ is a small number. After the short time $\epsilon$ the tail of the $Y$ arrow has been carried by along by the fluid to $x+\epsilon X$ and its head is wherever it has been carried to. One subtracts the flow-carried $Y$-arrow from the value of the vector field $Y(x+\epsilon X)$. The latter is represented by the small arrow whose tail is at $x+\epsilon X$ and whose head is at $(x+\epsilon X)+\eta Y(x+\epsilon X)$. Then you divide by $\epsilon$ and by $\eta$. The result is ${\mathcal L}_XY$.