There are many problems with this definition. For a start, plastic is an organic chemical, as is wood. A person is just as organic after they have died as they were when they were alive. So not everything that is organic is alive.
Carbon is a uniquely versatile atom in the number of long-chain chemicals it can form.
So it does make a good basis for life. There may also be more exotic possibilities, such as creatures made out of self-organising magnetic fields, or superconducting currents in the frigid regions of the comet cloud. One could, indeed, argue that there are some life-forms based on other elements here on Earth. Consider sand- dunes, for example. Each sand dune produced new sand dunes in its wake, as it deflects the air currents.