A course that my kids are doing in conic sections insists that all positions are represented as standard 2D cartesian coordinates $(x,y)$ (i.e. row vectors), and all translations are written as 2D column vectors $\begin{pmatrix}x\\y\end{pmatrix}$. Any deviation from this is marked incorrect.
Example:
I believe the reasoning is that positions are not vectors and so are written in row form, while translations are vectors and therefore must be written in column form.
As far as I can recall, I have seen positions and translations in either orientations according to whatever works in context, but usually both written as row vectors or both written as column vectors. If position and translation vectors are written with different orientations, translations are unable to be obtained from subtracting position vectors, and unable to be applied to positions by addition.
Is there any basis in theory or tradition for a strict separation of positions as row vectors and translations as column vectors?