You really should think of $\Delta$ as an $L^2$ self-adjoint, elliptic operator in it's own right irrespective of coordinates. $-\Delta$ has positive spectrum, with countable eigenvalues accumulating at $\infty$ etc. It's the average of the Hessian $\nabla^2 f$ in the sense that contracting the $(0, 2)$ tensor $\nabla^2 f$ with the metric $g$ produces a $(1, 1)$ tensor that you can trace (which is an averaging operation). In this sense there is no drift term.
On the other hand, there is a way think of the extra term $\mu$ coming from the connection. For a smooth function $f : M \to \mathbb{R}$, from the smooth structure alone, you can always form the differential $Df \in \Gamma^{\infty}(T^{\ast} M)$ as smooth section of the cotangent bundle. Without a connection, you can think of $Df : TM \to \mathbb{R}$ as a map between smooth manifolds, differentiate again, and this gives
$$
D^2 f : TTM \to \mathbb{R}
$$
I used $D$ for the differential so as not to confuse this with the exterior derivative $d$ which would have $d^2 = 0$.
Now if you have a connection, you can split $TTM$ as
$$
TTM \simeq VTM \oplus HTM
$$
into vertical and horizontal sub-bundles. In this particular case both $VTM$ and $HTM$ are isomorphic to $TM$. $VTM$ is the kernel of the map $d\pi : TTM \to TM$ where $\pi: TM \to M$ is the bundle projection and this is isomorphic to $TM$. The connection allows you to "split" the map $d\pi : TTM \to TM$ obtaining an injective bundle morphism $TM \to TTM$ complementary to $VTM \simeq TM$ whose image I'll denoted $HTM$. The vertical bundle consists of elements of $TTM$ that are tangent to the fibres of $TM$ while the horizontal bundles consists of elements of $TTM$ that are tangent to the base $M$.
To help clarify, for a general vector bundle $\pi : E \to M$, the kernel $VE$ of $d\pi : TE \to TM$ is isomorphic to $E$ and a connection gives a splitting $TM \to TE$ whose image is denoted $HE$ and such that $TE \simeq VE \oplus HE \simeq E \oplus TM$.
Now, what this all has to do with "drift" is that under the identification $TTM \simeq TM \oplus TM$, the $g^{ij} \partial_{ij}$ term comes from $VE$ - it's the term tangent to the fibre and corresponds to $\partial_{ij} f = D^2 f$. The other term - the one with the connection coefficients $\Gamma$ corresponds to the part that is tangent to the base $M$.
In other words, the $g^{ij} \partial_{ij}$ part arises from differentiating while moving along the fibres of $TM$ and the $\Gamma$ part arises from moving along the base $M$.
Thus the drift is measuring the change in the fibres of the bundle $TM$ as measured by the connection as the basepoint $x \in M$ varies.
