Actually, to be technically correct:
-it's not a game. the very last episode of Higurashi Kai makes a half-baked attempt at interactivity but it's only a guise to make you more invested in the final episode, the visual novels are purely linear, they're really novels that you read on your screen.
-People refer to the original as singular because it's a serial. The eight episodes form one story. If you wanna be really thorough Higurashi is considered as two novels though. First one is Higurashi no naku koro ni, made of the four question arcs, then there's Higurashi no naku koro ni kai, made of the four answer arcs (the anime's a bit different, because they didn't know if they'd be allowed to do a second season, so they crammed some of the answer arcs in the first). As such, each episode is always sold with the previous ones in the novel it's a part of, but the final whole is not one eight-episodes novel, it's two four-episodes novel sold separately.
Those two wholes are considered two novels in the "When they cry" series (yes, said series has an english title even in japanese). And then Umineko no naku koro ni (the four question arcs of Umineko) is "when they cry 3" and Umineko no naku koro ni rei (the four answer arcs of Umineko, only one published for now) is "when they cry 4". That also explains the odd titles of the english version somewhat.
That's for the PC versions though. As far as I understand on console they added interactivity and grouped all eight episodes in a single branching storyline (which you do have to play several times in different ways to eventually reach the good ending)