The latter two notations are the same if one takes
$\langle a,b \rangle :=\begin{pmatrix}
a\\
b
\end{pmatrix}$. This notation is sometimes used to have vectors written in-line. In this case, what you have is a linear combination of the two standard vectors:
$$x\langle 1,0 \rangle+y\langle 0,1 \rangle
=x\begin{pmatrix}
1\\
0
\end{pmatrix} + y\begin{pmatrix}
0\\
1
\end{pmatrix} =
\begin{pmatrix}
x\\
y
\end{pmatrix} = \langle x,y \rangle$$
Notice that these are both notations for vectors. The first notation is for a 2-by-2 matrix, which is not a vector unlike the latter two expressions. You can write the latter two expressions using the matrix-vector product, which might be the source of your confusion. In the 2-by-2 case the matrix-vector product is defined
$$\begin{bmatrix}
a &b \\
c & d
\end{bmatrix}\begin{pmatrix}
x\\
y
\end{pmatrix} =x\begin{pmatrix}
a\\
c
\end{pmatrix}+y\begin{pmatrix}
b\\
d
\end{pmatrix}$$
From this it should be trivial to see what 2-by-2 matrix is used to give you your latter two expressions.