It seems like there's a lot of things being mixed around in this question. First things first $\{(1,1),(2,2)\}$ is not a basis for $\mathbb{R}^2$; it is not linearly independent because $(2,2) = 2(1,1)$.
To put ourselves on firm footing,
Definition: Given some vector space $V$ with basis $\beta$ and $v \in V$ the coordinate vector for $v$ is $[v]_\beta = (a_1,\dots,a_n)$ where $a_i$ are the scalars needed to write $v$ as a linear combination of the basis vectors.
You write that in $\mathbb{R}^2$, any basis vector can be expressed as a linear combination of the standard basis vectors. This is technically correct, but I believe the source of your confusion. As you seem to understand, there are many different possible choices of basis for $\mathbb{R}^2$. The standard basis is $\{i,j\}$ where $i = (1,0),j=(0,1)$. Another possible example of a basis is $\{(1,0),(-2,1)\}$, and yes
\begin{align*}
(-1,0) & = 1i \\
(-2,1) & = -2i+j
\end{align*}
so that these vectors are a linear combination of the standard basis vectors.
But, it's the doubling up of the term basis vector that is unclear. Once you pick a basis for $\mathbb{R}^2$ you stick to it, and no longer would you refer to another set of vectors as the basis vectors. This part introduces the circular definition you are referring to.
With the definition of a coordinate vector and terminology I stated, let's revisit your example.
For example, consider a basis {a, b} where a = (2, 0) and b = (0, 2). The vectors of this basis can be rewritten as a = 2i + 0j and b = 0i + 2j. a and b are just coordinate vectors.
Issues:
- As I mentioned before this isn't a basis for $\mathbb{R}^2$.
- "The vectors of this basis can be written as $a = 2i+0j$ and $b = 0i+2j$" is awkward. You would simply say instead "The vectors $a,b$ may be written as $a = 2i+0j$ and $b=0i+2j$.
- You write immediately after "$a,b$ are just coordinate vectors." To talk about coordinate vectors, you must first fix a basis. Once you've picked a basis, it makes no sense to ask what the basis vectors are coordinates of if you've understood the definition of a coordinate vector.
To bring it all together, here's an example that brings together all the points.
Example: Let $V = \mathbb{R}^2$ and lets fix the basis $\beta = \{(1,0),(-2,1)\}$. Consider the vector $(3,2) \in V$.
- We may write $v$ as a linear combination of the basis vectors as
$$
(3,2) = 7(1,0)+2(-2,1).
$$
- The coordinate vector $[v]_\beta = (7,2)$ because these are the coefficients in the linear combination representing $v$.
If instead we took $\beta = \{i,j\}$ then for the same vector $(3,2)$ we have
$$
(3,2) = 3(1,0) +2(0,1) =3i+2j
$$
and the coordinate vector for $v$ is trivially itself.
