The safety release button , looking at your picture , is a ' restrictor ' hinge ( only opens so far before pushing that button ) , normally sold in pairs. you do get a little bit of movement from ' Friction stays ' ( window hinges ) when you wobble them.
Can you actually see where the hinge is broken ? ( Its unusual for hinges to break within 18 months )
Is the gap on the ' hinge ' side of the window ( closest to your wall ) ? ... you haven't yet answered this one asked by others, if not the glass could just need repfitting ( toe and heeling ) or as has been suggested the actual hinges can be moved on the sash/frame , it may be that the screws holding the friction stays to the sash have come loose ... I have seen this a few times!
If you are buying new hinges , buy 13mm stack with the stack height caps . Upvc has two different heights for the window hinges , 13mm or 17mm. Without taking a hinge off its very hard to measure the actual height , so some sell the smaller height with round packers to go below the hinge ( called stack height caps )
400mm side hung ( 16 inch ) restrictor stays are what you require from your description , IF the hinge is actually broken. Just bear in mind you will have to replace both hinges to ensure you get the correct ' angle ' of opening top and bottom , and the whole sash will need to be removed. You will either need someone to hold the sash whilst you unscrew it , or you will need to deglaze the sash to make it a lot lighter ( maybe worth doing this anyway to safe any potential glass breakages )
Honestly , if it were me , in your position , as much as you dislike the windowfitters, I would be getting them back to do it...It is under warranty, so why should you go to the hassle of paying out and doing it when they should be fixing it Free... Don't let the bad attitude and rudeness stop you getting what you are entitled to! Also if for any reason do you need to call them back in the future and you have repaired/replaced the hinges they may try and wiggle out and say you have invalidated any guarantee/warranty