The nonlinear coupling of two modes inside a laser, and in particular inside semiconductor lasers, is induced by the fact that each modes does not saturate only its own gain, but also the gain of other modes. Thus, the larger this cross-saturation effect, the stronger the competition between the modes. The intensity of this coupling depends on many things: the spevtroscopic details of the transition, the polarizations of the modes, their spatial overlap (especially in the case of standing wave leading to spatial hole burning), their frequency difference, etc...

Is this cross saturation effect absent in the gaseous gain medium lasers?

No the cross saturation effect also exists in gaseous gain media. But it can strongly depend on the frequency difference between the two waves because of spectral hole burning. See the chapter on inhomogeneously broadened lasers in my laser physics lecture notes, available at this address:

Thank you. But this document is not available.

Does mode competition and mode coupling refer to the same phenomena ?

There was a typo in the address. Here is the correct link:

http://www.lac.u-psud.fr/IMG/pdf/2015_laserphysics_bretenaker.pdf

I also attach it to this message.

And yes mode competition and mode coupling mean the same thing.