Torsion tensor: definition The definition of torsion tensor is the following:
$$
\mathbf{T}(\mathbf{X},\mathbf{Y})=\nabla_{\mathbf{X}}\mathbf{Y}-\nabla_{\mathbf{Y}}\mathbf{X} -\left[\mathbf{X},\mathbf{Y}\right]. 
$$
In an holonomic base $\left[\mathbf{e}_a,\mathbf{e}_b\right]=0$ the coordinates are
$$ 
T{^a_{\ \ bc}}e_a=\nabla_b e_c-\nabla_c e_b=(\Gamma^a_{\ cb}-\Gamma^a_{\ bc})e_a 
$$
due to $ \nabla_b\mathbf{e}_c=\Gamma^a_{\ \ cb}\mathbf{e}_a$. This result is different from the general result
$$
T^a_{\ \ bc}= \Gamma^a_{bc}-\Gamma^a_{cb}. 
$$ 

Where am I wrong in this calculations?

 A: It is just a matter of convention. MTW uses the convention that you have followed. Others, like Wald, use different one. Just a nuisance.
A: The connection (in this case the Levi-Civita conn.) $\Gamma^\alpha{}_\beta$ is a one-form. We can express it as (in a coordinate (holonomic) basis)
$$\Gamma^\alpha{}_\beta \equiv \Gamma_\mu{}^\alpha{}_\beta\, dx^\mu  \tag 1$$
on another hand in the case of the spin-connection we have
$$\omega^I{}_J \equiv \omega_\mu{}^I{}_J\, dx^\mu  \tag 2$$
There are two types of the indices on the above equations. I would call the indices that appearLHS of the equations only: $\{\alpha,\beta\}$ for eqn(1) and $\{ I,J \}$ for eqn (2) the vector space indices or transformation indices (not the standard words), and I would call the index $\mu$ on both eqn. the form index.
Consider an equation write with only the vector space indices
$$ \nabla {\bf e}^a = \Gamma^a{}_b {\bf e}_a \;.$$
There are two conventions to write in  term with the form index
CONVENTION 1
$$ \nabla_c {\bf e}^a = \Gamma_c{}^a{}_b {\bf e}_a =: \Gamma^a_{\underline{c}b}{\bf e}_a\;,$$
where the underline indicate that it is a  form index.
CONVENTION 2
$$ \nabla_c {\bf e}^a = \Gamma^a{}_{bc} {\bf e}_a =: \Gamma^a_{b \underline{c}}{\bf e}_a\;,$$
where the underline indicate that it is a  form index.
In the both case $c$  is a form index. 
If you notice that which index is a form index  you will never confuse.
Your second equation 
$$T{^a_{\ \ bc}}e_a=\nabla_b e_c-\nabla_c e_b=(\Gamma^a_{\ cb}-\Gamma^a_{\ bc})e_a \tag 3$$
is obviously use the CONVENTION 2 , we can write
$$T{^a_{\ \ bc}}e_a=\nabla_b e_c-\nabla_c e_b=(\Gamma^a_{\ c\underline b}-\Gamma^a_{\ b\underline c})e_a \tag 4$$
but your third equation us the CONVENTION 1 we can write it as
$$ T^a_{\ \ bc}= \Gamma^a_{\underline b c}-\Gamma^a_{\underline c b}. \tag 5 $$
$T^a_{\ \ bc}$ in (4) and (5) are the same.
