Reason for thinking of vector as "row" and "column" vectors in linear algebra Consider the $n$-tuple $(x_1,\ldots,x_n)$ with entries in some field $K$.
What is the reason for perceiving this tuple either as a row vector, $$ [x_1,\ldots,x_n]$$or as a column vector $$\left[\begin{array}{c}x_1\\\vdots\\x_n\end{array}\right]$$?
To clarify further: All the answers on this site, that I looked up, like this, this or this one deal with the question, which objects should be thought of as row respectively column vectors - opposed to that is what I'm asking: What's reason for making this distinction in the first place ?
Because the thing is, I could define matrix multiplication, like this $$ \left[\begin{array}{cc} a & b  \\ c & d   \end{array}\right](x_1,x_2):=(ax_1+bx_2,cx_1+dx_2), $$so I only ever need to deal with "tuple-vectors", not row or column vectors; no need to ever talk about row or column vectors. So the thing is, that everything related to coordinates could be done, in the case of vectors, in row or column terms. So if that is possible, why is nobody doing it like that ? 
The only benefit that I see from making the row-column distinction - in contrast to using tuples $(x_1,\ldots,x_n)$ which are neither "row" nor "column"-type - is to be gained on a notational level, so that it is easier to remember, for example, how to do matrix-vector multiplication, by "moving" along the rows of the matrix while "moving" along the  column of vector.
But that seems a shallow reason, to put up with distinguishing between "row" and "column"-vectors all the time. I hope there's something deeper than that.
 A: First, there is usually not made any formal difference between the two points of view: there is only one notation $K^n$ for the space of $n$-tuples of scalars, not separate ones for those tuples written vertically or horizontally (or maybe in other forms still).
However, once one uses matrices to represent linear maps, one has to make the choice whether to organise them so that the coordinates (at arrival) of the images of the vectors of the source basis used are given by its columns or by its rows. The most common convention is to have the images correspond to the columns of the matrix, but some authors (in my experience mostly those who do not rely heavily on matrix calculations in the first place) use the convention to use the rows of the matrix. Supposing columns are used, the effect of a linear map on the coordinates of the vectors it operates on is given by left-multpication by the matrix of the linear map, and that operation requires the coordinates of the vector acted upon to be written as a column vector in the matrix product. So of $A$ is the matrix, with respect to chosen bases, of a linear map $f:V\to W$, and $v\in V$ has coordinates $(v_1,\ldots,v_n)$ on the chosen basis of $V$, then the coordinates of $f(v)$ in the chosen basis of$~W$ are the components of the product
$$
  A \cdot \begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}.
$$
If we had written the coordinates as a row vector, then the only way to multiply by a matrix is to do so on the right (and by the transpose of $A$ if it is not square), but that simply does not give the right coordinates of $f(v)$. So writing column vectors is really just an easy way to remember what convention was used to represent linear maps by matrices.
Although at almost every point where a choice is made there are some authors that choose the opposite convention (and this even applies to the question whether the row or column index in a matrix is written first), I think that fortunately everybody agrees about the rule to multiply matrices (taking rows on the left multiplied by columns on the right). The above is based on the assumption that this convention is universal. Changing that convention would be a most certain way to be misunderstood by everyone. Moreover if one would define matrix mutliplication in a way consistent which what you write in the question, then as user68061 observed you loose associativity of multiplication, which means that matrix multiplication no longer corresponds to (linear) function composition, and it in fact become a completely pointless operation.
A: If you just have a tuple of numbers then, as you said, there is no difference between column and row. However, if you want to have nice matrix algebra, you have to distinguish between two.
The reason nice matrix algebra exists is the fact that, any $m \times n$ matrix can be understood as a linear map from $K^n$ to $K^m$. In this formalism, a column vector is a map from $K$ to $K^n$ i.e., a vector; but a row vector is a map from $K^n$ to $K$ i.e., a functional.  
If you define the multiplication as you did, you will lose associativity, and that is bad.
