If you do not mind a used / refurbished laptop, then I recommend you consider the following Dll Latitude 3540 with a core i5-4200u, Radeon HD 8850m and 1080p screen for $175 + shipping. The seller has a 100% feedback from over 8,000 feedback.
I actually purchased a refurbished Dell Latitude 3540 back in Jan 2014 for about $500 (different seller), and I am pretty satisfied with my purchase. The Radeon HD 8850m was considered to be a good mid range gaming GPU back in 2013 / 2014. It's performance is basically equal to the nVidia 945m which means it is a little more powerful than the common nVidia 940mx.
With the release of more powerful GPUs and more graphically demanding games, the Radeon HD 8850m is currently considered to be a budget GPU. But this more than enough for a basic game like Minecraft. However, recording gameplay is pretty demanding on the CPU. The i5-4200u is only a dual core CPU so you will definitely notice lower game performance when recording game play.
If you want a new laptop for $200, then you are going to have to settle for something that is much less powerful since only laptops with low end CPUs sells for such a low price. Probably one of the "best" new $200 laptop is the Acer Aspire 1 because it is the only laptop with a low end Intel quad core Celeron N3450 CPU selling for $200. Other laptops with that low end CPU are typically at least $330 based on Amazon prices. But they have a 500GB hard drive rather than only 32GB of eMMC storage (faster than a hard drive, but much less slower than a SSD). 32GB of storage is definitely not very much. With basically only Windows 10 installed and a few basic programs (Firefox, antivirus, printer drivers, etc) installed on the C: drive of my two laptops, Windows is basically using about 25GB of storage.
Here's a link to the Acer Aspire 1 on Amazon and a video review. Keep your expectations low and do not expect to be able to record game play with it while playing Minecraft and still have ok frame rates.