That's exactly what my loo flusher is Jon, a 'push' one. And the photo shows the part my plumber changed for me. It was weird having no water whatsoever all evening and all night. Made me appreciate all the more having clean water on tap.
When I was a little girl, everyone had a cistern up on the wall and a chain to pull. (remember those?) I got a smack for yanking it far too hard and nearly pulling the whole thing down.
Makes my husband's 'toilet arrangements' in Africa look a bit more sensible (Hole in the ground and a pot of water to wash one's bum - no toilet flushing and no toilet paper!)