Did my teacher specialist about sulfur compounds made an error in part of Ramberg-Bäcklund rearragement mechanism?
As discussed above, the pathway shown on the left is incorrect because it essentially shows insertion into a $\ce{C-H}$ bond by an anion with loss of $\ce{X^-}$, which is not the easiest path for this reaction to take by a long shot.

The actual mechanism for this reaction is as shown below.


First, the non-halogenated alpha-carbon is deprotonated.
The carbanion formed arranges itself such that the it can attack the halogenated carbon and in such a way that the two R-groups are cis to each other.
Attack occurs, with loss of $\ce{X^-}$. Product is cis.
In water, due to lack of solubility, decomposition is faster than redeprotonation and the (Z)-alkene forms.
In organic solvent, base deprotonates an alpha-carbon, which can then scramble.
The main reprotonation product is the trans product.
The trans product decomposes, and the (E)-alkene is the major product.




A few notes:


The red path makes almost completely (Z) product. The blue path makes a mix of the two with the (E) product as the main product.
Purple star: the reason the anion attacks to form the cis ring is unknown, but there are several theories, apparently.
Green stars: cheletropic ring decomposition is symmetry forbidden, so this is theorized to be a two-step decomposition through either diradical bond fissions or a dipolar mechanism.