Beam Spllitters What is the difference between a symmetric, balanced and an asymmetric beam splitter? Which beam splitter should I use if I want one input state to be fully transmitted and the other to be fully reflected i.e. if I want both of the inputs to end in the same output channel?
 A: A balanced beam splitter provides a 50% probability for the light from each path to end in each of the two output paths. An asymmetric beam splitter provides different probabilities. I do not know of a distinction between balanced and symmetric beam splitters, but I can envision some authors caring about the relative phases that the light picks up by reflecting versus being transmitted at a beam splitter.
Your desired beam splitter seems physically impossible. If we consider the inverse problem, your condition means that light input from one of the two output ports would not be transmitted nor reflected, which does not work. Perhaps it can work if you supplement the beam splitter with an absorbing barrier / beam dump just past the undesired output port.
In math: the beam splitter transformation goes like
$$\begin{bmatrix} E_c \\ E_d \end{bmatrix} =
\begin{bmatrix} \sqrt{R}e^{i\phi_{ac}}& \sqrt{T}e^{i\phi_{bc}} \\  \sqrt{T}e^{i\phi_{ad}}& \sqrt{R}e^{i\phi_{bd}} \end{bmatrix}
\begin{bmatrix} E_a \\ E_b \end{bmatrix},$$ where $R$ is the reflection probability and $T$ is the transmission probability, subject to $R+T=1$ and $\phi_{ad}+\phi_{bc}-\phi_{bd}-\phi_{ac}=\pi$. For there to be no output in one port, both $R$ and $T$ would have to be zero, which does not describe a beam splitter. For there to be equal probability of going in each direction, we need $R=T=1/2$. To be fully symmetric, we can choose something like $\phi_{bd}=\phi_{ac}=0$ and $\phi_{ad}+\phi_{bc}=\pi/2$.
