I would say that you should open a ticket to send your business details including a website to us, so we may ask the different banks in Cyprus if they will accept your business specific. The simple and broad answer to your question is YES all banks will accept this, but this does not mean that they will accept your business case, since no business case is the same.
There is a difference in Games and Gaming websites, for instant Gaming is like Poker, Online Casino and so on, Games are like second life and such like.