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.
In one sense, you can say that a vector is simply an object with certain
properties, and it is neither a row of numbers nor a column of numbers.
But in practice, we often want to use a list of $n$ numeric coordinates to describe
an $n$-dimensional vector, and we call this list of coordinates a vector.
The general convention seems to be that the coordinates are listed in the
format known as a column vector, which is (or at least, which acts like)
an $n \times 1$ matrix.
This has the nice property that if $v$ is a vector and $M$ is a matrix
representing a linear transformation, the product $Mx$, computed by the usual
rules of matrix multiplication, is another vector (specifically, a column vector)
representing the image of $v$ under that transformation.
But because we write mostly in a horizontal direction and it is not always
convenient to list the coordinates of a vector from left to right.
If you're careful, you might write
$$ \langle x_1, x_2, \ldots, x_n \rangle^T $$
meaning the transpose of the row vector $\langle x_1, x_2, \ldots, x_n \rangle$;
that is, we want the convenience of left-to-right notation but we
make it clear that we actually mean a column vector
(which is what you get when you transpose a row vector).
If we're not being careful, however, we might just write
$\langle x_1, x_2, \ldots, x_n \rangle$
as our "vector" and assume everyone will understand what we mean.
Occasionally we actually need the coordinates of a vector in row-vector format,
in which case we can represent that by transposing a column vector.
For example, if $u$ and $v$ are vectors (that is, column vectors), then the
usual inner product of $u$ and $v$ can be written $u^T v$, evaluated as
the product of a $1\times n$ matrix with an $n \times 1$ matrix.
Note that if $u$ is a (column) vector, then $u^T$
really is a row vector and can (and should) legitimately be written as
$\langle u_1, u_2, \ldots, u_n \rangle$.
This all works out quite neatly and conveniently when people are careful
and precise in how they write things.
At a deeper and more abstract level you can formalize these ideas as shown in
another answer.
(My answer here is relatively informal, intended merely to give a sense of why
people think of the column vector as "the" representation of an abstract vector.)
When people are not careful and precise it may help to say to yourself sometimes
that the transpose of a certain vector representation is intended in a
certain context even though the person writing that representation
neglected to indicate it.
Best Answer
Let $n$ be a positive integer.
Of course we identify lists, row and column vectors through "obvious" bijections. For example, given a list$$(x_1,\ldots,x_n)\in\mathbb R^n$$ we can set $$\forall i:x_{i,1}:=x_i$$and $$\begin{pmatrix}x_{1,1}\\\vdots\\x_{n,1}\end{pmatrix}\in\mathbb R^{n\times 1}$$is the column vector associated to the list.