Find plaintext from two ciphers encrypted with a one-time pad and the same key
First, there seems to be a misunderstanding: The one-time pad is the key. So your equations simplify to
$$ C_1 = L_1 + K \text, $$ 
$$ C_2 = L_2 + K \text. $$

At this point, you would be right if the $L_i$ could contain arbitrary bit sequences. However, the question imposes a constraint: $L_i\in\{A=\mathtt{1111},B=\mathtt{0011},C=\mathtt{1010}\}$.

Let's look only at the first bit of all the variables (I'll denote these in lower-case). Since $c_1=c_2$ by observation, you know that $l_1=l_2$ from the equations above. Hence given the alphabet from the question, we know that either both $L_1$ and $L_2$ are $B=\mathtt{0011}$, or they are both not $B$. You can proceed similarly for the next bits, and at some point you should be able to reduce the set of possibilities to one unique solution.