Meaning and application of the connection coefficients (Christoffel symbols)

You are using two different sets of basis vectors. $frac{partialmathbf e_r}{partial theta} = mathbf e_theta$ and $frac{partial mathbf e_theta}{partial theta} = -mathbf e_r$ hold for the orthonormal polar basis, in which the metric takes the form

$$g_{ij} = pmatrix{1 & 0 0 & 1}$$

The connection coefficients you quote arise from the polar coordinate basis $left{frac{partial}{partial r},frac{partial}{partial theta}right}$ which is not orthonormal, and in which the metric takes the form

$$g_{ij} = pmatrix{1 & 0 0 & r^2}$$

The two bases are related via $mathbf e_r = frac{partial}{partial r}$ and $mathbf e_theta = rfrac{partial }{partial theta}$.

---

An important thing to recognize is that the basis $left{frac{partial}{partial r},frac{partial}{partial theta}right}$ arises naturally as the basis induced by the polar coordinates $(r,theta)$. On the other hand, the orthonormal basis ${e_r,e_theta}$ is not induced by a coordinate system; there is no set of coordinates $(u,v)$ such that $e_r = frac{partial}{partial u}$ and $e_theta =frac{partial}{partial v}$. This is an example of a non-holonomic basis.

The reason that this is important is that in your first pass through GR, you will likely start off by using holonomic bases exclusively. Accidentally using a non-holonomic basis can lead to some apparent contradictions.