Combinatorial identity from squaring the binomial expansion
Your equation (2) can be rewritten as $$\sum_{k=1}^{m-1} C_{k-1}C_{m-k-1} = C_{m-1},$$ where $C_n$ is the $n^{\text{th}}$ Catalan number. Any introductory article on Catalan numbers should contain one or more counting proofs of that identity.