Look for local companies that buy used corporate laptops and refurbish them. Lots of barely used Lenovo, Dell etc. computers.
You can get Lenovo x1 carbon pretty cheap with pretty high end cpu on ebay. Don't know if it can handle Minecraft though. Absolutely nothing with a discreet graphics card will fall into your price range. I would recommend your relative keep saving until they can spend a little more.
It really doesn't need a discreet graphics card I myself have a Lenovo ThinkPad L540 which has pretty good performance and can handle minecraft good enough, but my relative really dislikes how thick and bulky it is. Is there maybe a thinner laptop in that range with similar performance?
Absolutely not. The carbon is already very slim. It really sounds like they want a MacBook which we both know is not gonna happen for 200. Your relative needs to come to grips with the reality of their budget or just keep saving. Best time to buy is after Christmas when people get new shit and sell their old stuff.
One of the requirements is at least not going to be it. Old secondhand ThinkPads are a good option. To my Knowledge Minecraft in Linux was kinda a hassle to get working properly. But a old w530 for example is a good laptop that might be what you want but they are not ultra slim about 2cm thick
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