Hey you ;)
I'm pretty sure I got the concept of influenced/triggered heats/ruts for betas from this fic: https://archiveofourown.org/works/31099799/chapters/76839263. I would basically mean that as a beta Minho could go into heat if the alpha he's with goes into rut, and if he's with an omega in heat he would go into rut. Of course there would also be some sort of suppressants / pheromone blockers to stop that from happening. I'm still thinking the concept through myself, but I think it would need extended exposure / bonds to trigger a heat/rut in a beta like this, since otherwise betas would be susceptible to every rutting alpha in the world and that just wouldn't make sense.

Also, would it mean that after having induced ruts (as a consequence of being w chan( he would have a problem being around the omegas, opposed to before?
Nah, it would just mean that Minho can't keep telling himself that Chan is not special to him ;)