To me, the most general would be a Memory Palace, all other three would be Memory Palaces but a special kind.
A Memory Palace to me is just what it is to everyone here I guess, a visualized mental location where information is stored in a way that makes it easy/easier to remember. It can be anything from a building (like the one most of us start with), pictures from a book, scenes from a movie, whatever you use to attach information to.
Roman Rooms to me are the simplest kinds of Memory Palaces, ones that resemble the room in Simonides’ memory. It can be multiple rooms, but it doesn’t have to be filled with predefined loci or a certain route. You can mentally walk around in it without having a certain direction. Like with Simonides, this includes rooms on places like birthdays, where you remember everyones place in the room. Also when making a memory journey, it starts with a roman room, in which you define loci and a route. Most of my Roman Rooms do have predifined loci, but no predifined journey.
As mentioned before, Memory Journeys have predifined loci which are walked though in a predifined order, a Journey. These Memory Palaces I use for things when an actual order is important, not much else to be told about that.
A Mind Palace is a Memory Palace, though not specifically for memories (but it can have some). These are the more vivid places, I can go to them and totally lose the outer world. I usually do so when I am stressed or emotional. I only have one of these, and when I am there, it has a lot of things that calm me down. I hear my favorite piano piece play, I see my favorite stuffed animal of all time (which I also still have IRL), my daughter is there to hug me, and basically I just relax there and let all stress flow away. It is almost as vivid as the real world, and when inside, I can’t do anything on the outer world as my full attention is focussed inward.
When rewatching Sherlock, I found that my mind palace is actually very similar to his. I can go there, think, relax, etc. Though I don’t need too much stimuli from the outside (Like his line, “Get out, I need to go to my mind palace”).