Why augmenting the training data with binary attributes works better for our dataset?
Your third model has the most capacity of the three.  When you have the values $x$ with special negative value $\xi$  and indicators $\mathbf{1}_0$ and $\mathbf{1}_{\xi}$, the linear model can fit
$$ \alpha x + \beta \mathbf1_0 + \gamma \mathbf1_{\xi}$$ 
When $x&gt;0$ you just get $\alpha x$, when $x=0$ you just get $\beta$, and when $x=\xi$ you get $\alpha \xi + \gamma$; 
so your model can get the correct (linear) signal out of the positive values and arbitrary values for the two special values.