Actually, it hasn't been explained very well to you. There aren't different names for the same thing.
USB-C is a port type. It can be a regular USB-C port, or it can use the Thunderbolt (usually Thunderbolt 3) *protocol*.
mini-Displayport is a port type. It can be a plain mini-Displayport (used exclusively for video connections) or it can use the Thunderbolt (usually Thunderbolt 2) *protocol*.
Any port that uses the Thunderbolt protocol is significantly faster than the same port that doesn't use the Thunderbolt protocol.
I originally suggested that you go with a 1080p monitor that I recommended because 1) I doubt that you will even notice that it is a lower resolution than a 4K monitor, 2) a 1080p monitor should be cheaper (but the one that I recommended should still look fantastic), 3) using an HDMI connection should be more straightforward and less confusing (assuming that you get a high quality HDMI cable, which I gave a link for), 4) I think that if you try and run a 4K monitor with your 2015 Mac mini you will be disappointed with the results. 2015 minis didn't have particularly powerful graphics cards.
You are listening to a lot of cooks right now trying to help you make the same stew. I won't try to confuse you further.
Not at all. Every little bit slowly gets me closer to the entire picture. Muchly appreciated!