"Lovecraftian game" is an improbable label to begin with. We don't say "Conan-Doylian game" or "Gibsonian" or "George-Lucasian". Games have adapted stories by Poe and Verne and so on, but not enough to talk about them as a category. Tolkien adaptations are an industry in themselves, but you'd say "Middle Earth games" -- maybe -- certainly not "Tolkienian".
Lovecraft, as always, is weirdly liminal: a collection of tropes that can be, and arguably should be, detached from the stories themselves.
So. A game genre (subgenre) is a set of conventions about what you do. Except Lovecraftian-ness isn't defined by what you do; it's defined by what happens to you. You are shaken out of your mundane reality and forced to confront the cosmically terrifying. (Or cosmically astonishing, but we'll get there.) What you do in your mundane reality is an open question. The opening and closing of the story, before and after revelation, can go anywhere! It's not like a "Sherlock Holmes" game where you know exactly what's on the menu.
Of course, a game that leads up to cosmic discoveries may well start out with mundane discoveries. That's an obvious model; it's why the Call of Cthulhu RPG is canonically about investigators. I don't have hard stats, but I'd bet that the most common Lovecraftian crossover is the hard-headed private eye who's seen it all and is about to see a whole lot more.
Now look: I'm going to ignore the kind of game where phallic oozing horrors swarm at you and you mow them down with a shotgun. That's Lovecraftian only in the sense of H. R. Giger and the Aliens franchise -- the influence is unarguable, but anything you can blow up ain't cosmic. (Nor run over with a boat; really, don't get me started.)
(Yes, I'm handwaving a lot of ground here. The FPS Gigeresques led directly to survival horror by simply deleting the shotgun. Many undeniably Lovecraftian titles there. But "shooting" and "dodging" are the easy answers in game design, right? And even with those answers, players demand at least a little plot.)
A lot of Lovecraftian games start out with the general question: what's going on here? As these games indeed do. But that doesn't mean that you do the same stuff in them. Let's compare.
- Moons of Madness
- Old Gods Rising
- Call of the Sea
- Paradise Killer