Plumbing washing machine waste drain hose

Hi, can I please get some advice?
I have just moved into a house where the laundry tub does not have a bypass hole in the sink. We have just had it draining into the sink, but it has accidentally overflowed a few times so I want to plumb it in.

Other than buying a new sink with a waste bypass kit, how can I do this? To me it looks like the waste connection is for a dishwasher hose only - is that right? If I want to connect the washing machine waste, do I need to buy a new p-trap similar to the one in this link? :https://i.ebayimg.com/images/g/YPIAAOSwrhBZEvuu/s-l400.jpg

Will I even be able to connect the waste hose if it is rigid plastic? Or does it need to be rubber in order to connect to a waste outlet on a p trap?

Or am I better off replacing the sink with one with a bypass? Is that a better outcome, or same same?

Photos attached
Thanks!

Our washing machine drain hose connects to the "dishwasher outlet". Does your hose not have a rubber end? If not, perhaps you could contact the washing machine manufacture to see if they have an available solution.

Ditto, as per the poster above, mine just connects to the barbed fitting as shown.