Love all the different names for the same thing. So Verrry helpful. NOT!
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.