I will explain why your first picture is exactly what you need for your program, it takes a very slight bit of algebra and definitions to write it out, for instance, given two vectors $\vec{a} = (a_1,a_2,a_3,\dots, a_n) $ and $\vec{b} = (b_1,b_2,b_3,\dots,b_n)$, then:
$$\langle \vec{a}, \vec{b} \rangle \equiv \sum_i a_i b_i = a_1b_1 + a_2b_2 + \dots a_nb_n $$
When the field of scalars used is $\mathbb{R}$ this product is bilinear in its arguments, that is
$$\langle a + c, b \rangle = \langle a, b \rangle + \langle c, b \rangle$$
and
$$\langle a , b + c \rangle = \langle a, b \rangle + \langle a, c \rangle$$
hence
$$\langle a + c, b+d \rangle = \langle a, b \rangle + \langle a, d \rangle + \langle c, b \rangle + \langle c, d \rangle$$
Additionally
$$||\vec{a}|| \equiv \sqrt{\vec{a} \cdot \vec{a}} \rightarrow ||\vec{a}||^2 = \vec{a} \cdot \vec{a} = \langle \vec{a}, \vec{a} \rangle $$
The first picture contains such objects, lets start with the first one ($M=m_1 + m_2$):
\begin{eqnarray*}
\vec{v}_1' &=& \vec{v}_1 - \frac{2m_2}{M}\frac{\langle\vec{v}_1 - \vec{v}_2, \vec{x}_1 - \vec{x}_2 \rangle}{||\vec{x}_2 - \vec{x}_1 ||^2}(\vec{x}_2 - \vec{x}_1) = \vec{v}_1 - \frac{2m_2}{M}\frac{\langle\vec{v}_1 - \vec{v}_2, \vec{x}_1 - \vec{x}_2 \rangle}{\langle \vec{x}_2 - \vec{x}_1,\vec{x}_2 - \vec{x}_1 \rangle}(\vec{x}_2 - \vec{x}_1) \\
&=& \vec{v}_1 - \frac{2m_2}{M}\frac{(\vec{v}_1\cdot\vec{x}_1 - \vec{v}_1 \cdot \vec{x}_2 - \vec{v}_2 \cdot \vec{x}_1 + \vec{v}_2\cdot \vec{x}_2)}{(\vec{x}_2\cdot\vec{x}_2 - \vec{x}_2 \cdot \vec{x}_1 - \vec{x}_1 \cdot \vec{x}_2 + \vec{x}_1\cdot\vec{x}_1)}(\vec{x}_2 - \vec{x}_1) \\
&=& \vec{v}_1 - \frac{2m_2}{M}\frac{(v_{1x}x_{1x} + v_{1y}x_{1y} - v_{1x}x_{2x} - v_{1y}x_{2y} - v_{2x}x_{1x} - v_{2y}x_{1y} + v_{2x}x_{2x} + v_{2y}x_{y})}{(v_{1x}x_{1x} + v_{1y}x_{1y} - v_{1x}x_{2x} - v_{1y}x_{2y} - v_{2x}x_{1x} - v_{2y}x_{1y} + v_{2x}x_{2x} + v_{2y}x_{y})}(\vec{x}_2 - \vec{x}_1)
\end{eqnarray*}
Those large pieces are all in terms of the given initial components of the velocities and positions where I have used subscripts to denote the first and second vectors x and y components. This single equation is also two equations in one for the x and y components of $\vec{v}_1'$.
$$v_{1x}' = v_{1x} - \frac{2m_2}{M}\frac{(v_{1x}x_{1x} + v_{1y}x_{1y} - v_{1x}x_{2x} - v_{1y}x_{2y} - v_{2x}x_{1x} - v_{2y}x_{1y} + v_{2x}x_{2x} + v_{2y}x_{y})}{(v_{1x}x_{1x} + v_{1y}x_{1y} - v_{1x}x_{2x} - v_{1y}x_{2y} - v_{2x}x_{1x} - v_{2y}x_{1y} + v_{2x}x_{2x} + v_{2y}x_{y})}(x_{2x} - x_{1x}) $$
And
$$v_{1y}' = v_{1y} - \frac{2m_2}{M}\frac{(v_{1x}x_{1x} + v_{1y}x_{1y} - v_{1x}x_{2x} - v_{1y}x_{2y} - v_{2x}x_{1x} - v_{2y}x_{1y} + v_{2x}x_{2x} + v_{2y}x_{y})}{(v_{1x}x_{1x} + v_{1y}x_{1y} - v_{1x}x_{2x} - v_{1y}x_{2y} - v_{2x}x_{1x} - v_{2y}x_{1y} + v_{2x}x_{2x} + v_{2y}x_{y})}(x_{2y} - x_{1y}) $$
As can be seen this is entirely without any mention at all of the angle of impact, and is solely in terms of the initial components. A similar argument can be applied to the second equation to obtain the four equations for the components of the final velocity.
