Overall, You don't need a new or recent system to run game servers. an older system is usually quite capable of a reasonable server experience. I've used my T3200 based Dual Core laptop for running a Minecraft server back when I worked on GriefPrevention for testing changes, and it worked well enough. I used my Q8200 based Quad Core system for running the actual build server for our entire company and it kept up admirably, and that was far more load than it would ever see as a Game Server.
if you want to do a separate build for it, you likely don't need the latest processors. The latest Xeons and i7's are pricey but you could go back a few generations. If you go back far enough you can find Xeons and even Dual-Socket motherboards for cheap. (But don't go back too far because while they get cheaper you are suddenly in single-core territory, like with Socket 604)
For example, there are Haswell-based Xeon chips available for only about 20-30 dollars more than the i3 you've listed. UNlike the i3 though you might be able to find an affordable dual-socket motherboard for it as well, specifically for use as a server.
For RAM, if you are running all of those you would definitely want more. This goes particularly if the Minecraft server is "modded" in some way. I use -Xmx8G just for my Single Player Modded setup for example, just to be safe.
Otherwise, If you are only running one at a time and switching, then you could probably do fine with 16GB. But of course then t here is shutting down the old game, starting the new one, etc. and if the server is headless you need to do that through RDP and it's a pain.
