T. Koshy, *Catalan Numbers with Applications* (Oxford, 2009), page 322, proves a very similar identity:
$$C_n=\sum_{k=1}^{\left\lfloor\frac{n+1}{2}\right\rfloor}(-1)^{k-1} \binom{n-k+1}{k} C_{n-k}$$
$$\Leftrightarrow \sum_{k=1}^{\left\lfloor\frac{n+1}{2}\right\rfloor} (-1)^{k-1} \binom{n-k+1}{k} C_{n-k}=0.$$
(I did not check the book, the identity is quoted in this paper, equation 4.)

I also note that Mathematica knows the identity in the OP, in the form

$$\sum_{k=0}^{\left\lceil\frac{n}{2} \right\rceil
} (-1)^{k+1} \binom{n-k}{k} C_{n-k}=-1,$$
(Floor or ceiling in the summation limits does not matter, the sum may also be extended to infinity, since the binomial coefficients vanish.)

In response to a question in the comment, I record the q-analog of Koshy's identity, derived by George Andrews.

(The sum over $r$ may be terminated at $\left\lfloor\frac{n+1}{2}\right\rfloor$.)