Consider a fibre bundle $E$ and a certain connection, $TE=H\oplus V$. A path $\gamma(t)\in B$ can be horizontally lifted to a path $\gamma(t)\in E$ according to $\pi\circ\tilde\gamma=\gamma$ and $\tilde\gamma'\in H$. Apparently, this lift is unique.
I seem to be having a bit of a mental block. Why is this lift unique? Consider e.g. a trivial bundle $E=B\times F$, with connection $H=TB\times\{0\}$. Then for any $f\in F$, the lift $\tilde\gamma(t):=(\gamma(t),f)$ is a horizontal lift of $\gamma(t)\in B$, right? So how is this unique? If any $f$ defines an horizontal lift, it most certainly is not unique. What gives?
When one wants to lift a homotopy $f:X\times [0,1]\to B$ to the total space, part of the data always is a lift $\tilde f_0: X\times \{0\}\to E$ so that $\pi\circ \tilde f_0= f_0$. With this in mind it may be clear why when one says a lift is unique, one means unique up to the choice of this initial "lift at zero".
Specifically the theorem should be reformulated here as: For any point $f$ in the fibre of $\pi^{-1}(\gamma(0))$ there is a unique horizontal lift $\tilde\gamma$ of $\gamma$ so that $\tilde\gamma(0)=f$ and $\pi\circ\tilde\gamma = \gamma$.