Stuck plug in sink, please help

We just moved into a new house, and the sink in the upstairs toilet looked to be draining slow, so I tried to move the plug. It has a metal plug attached to the sink, and I doesn't have a lever so thought it was the push in, push again to pop out kind. Now it's stuck in the drain and don't have enough space to get something down the side to pry it back out.

The other two photos are of the downstairs sink which I'm guessing is the same type of plug, and that doesn't want to push down at all. We are in a very hard water area so it's possible any mechanisms are gummed up with limescale, but the house is only about 6 years old.

So, is there an easy way to get the plug out? And was I right about it supposed to be on a spring type?