An RTS is Real-Time, obviously, and has both base building and unit control. For example:
Command & Conquer
Warcraft
Starcraft
Total Annihilation
Warhammer 40K: Dawn of War
Age of Empires
I've seen many games listed in here that just have the unit control aspect, which is typically referred to as RTT (Real-Time Tactics). For example:
MechCommander
Warhammer 40: Dawn of War II
Sudden Strike
Total War series (technically a combination of Grand Strategy with RTT battles)
Both of these are great genres, but can appeal to different audiences because they feel and play completely different (personally, I enjoy them both). Unfortunately, developers and publishers started to call RTT RTS a number of years back and effectively erased the RTT sub-genre, muddying the waters and making it difficult for players to find the games they want to play. It doesn't help that they still call MOBAs RTS, or worse, when Turn-Based Strategy games are labelled RTS (like, really, Panzer Genreal is RTS?).
As for Dungeon Keeper, while technically an RTS, it has base building and (limited) unit control, I consider it and other games like it (War for the Overworld, Evil Genius, Dungeons, etc) their own sub-genre, since the theme and feel of the game is quite different from your typical RTS. If I'm looking for a game like Dungeon Keeper, I'm typically not looking for something like Age of Empires, or MechCommander, or RimWorld, or Crusader Kings II, or Defense Grid.
While all of the games listed above are both Real-Time and contain elements of Strategy, using the sub-genres help players find the games they want to play. It would be like lumping WoW, FFVII, Diablo, Grimrock, and Disgaea all together, when in fact they are all different sub-genres of "RPG" (MMORPG, JRPG, ARPG, Dungeon Crawler, TRPG).
All that said, outside of Blizzard, I would say that the traditional RTS is not in a good place. I think part of this is because of the above muddying, and players are tired of sifting through myriad of games that are labelled RTS but aren't RTS. The other part is because RTS went competitive for a while (StarCraft) and developers shifted to try and get into that same spotlight, not realizing that they weren't Blizzard. There were plenty of great RTS games before StarCraft, just as there were great MMOs before WoW, but people wanted to imitate instead of innovate, and that really hurt the genre, IMO. Is it a dead genre? No, there are still a few RTS games coming out from time to time, like Grey Goo, Northguard, maybe They are Billions (I'm torn between calling it RTS or Tower Defense, its kind of a mix of both so maybe it still counts).