It’s best to think of the Gates of Hell trilogy as something unofficial, rather than a planned thing. Catriona MacColl is in all three films, under three differently-spelled version of her name. She’s the heroine in each, but a different character each time, but always fighting creatures from hell. One would like to think Fulci planned this to create some sort of through line, but’s far more likely just because she was available.
The other two films in the trilogy are absolutely sane, compared to House By the Cemetery. I’ve seen it half a dozen times and read the plot summary on Wikipedia, and I still have no idea what this is all about. It doesn’t help that I have a pretty rough copy of the film, likely edited all to shit. However, I’ve little faith that a better DVD release would make even the slightest improvement.