How to distinguish between the different frequency domains?
This answer is primarily opinion. It seems to me that the most sensible thing to do is to make the transform of interest clear in your text. Then you can use "frequency domain" exclusively and unambiguously. 

For Laplace transforms, "s domain" will work as well, and I do believe they are interchangeable. I do not believe "Fourier domain" would be correct usage in this case. "s domain" may be preferred if you wish to emphasize that real values of $s$ have a clear aperiodic interpretation (exponentials in the time domain).

If the transform is not yet clear, you could say "Fourier domain", I suppose. In that case the assumed spectrum would be the complex (possibly conjugate symmetric) form.

Still, if it is within your control, be explicit about the chosen transform.