The wikipedia definition is a fine definition that you can use for your paper if you need one but I think you're missing something.
The $\epsilon$ is random error, which is synonymous with noise. In practice, the random error can be Gaussian distributed, in which case it is Gaussian noise, but it could take on other distributions. If the distribution of $\epsilon$ happens to be Gaussian then you've met one of the theoretical assumptions of the model and things like interval estimation are better justified. If it's not Gaussian then, like Glen_b said, you still have that it's best linear unbiased.
Theoretically, the random error (noise) is supposed to be Gaussian distributed but the outcome could be anything. So, in order to answer your question you'd need to state whether you want to know the distribution of your particular noise or what the distribution of the noise should be. For the former you'd need data.