While I do agree with most of the points above, I don't think a horror game meant to be psychologically scary has any real need to make sense. Reason is one of the factors that helps us rationalise things, not to mention that humans are afraid of the unknown (extremely so). One of the great things about focusing on cult's is that, generally, what they do does not make any sense. Beyond the whole 'sacrifice lesser beings to summon/appease greater beings', most sacrificial cults just do not make sense at all.
I noticed how you said you wanted to be 'somewhat' authentic. To be honest, occultism isn't widely known, or even thought of, beyond the sacrificing. People only see a cow getting massacred and say 'wow, that's barbaric' which is ironic but that's going off-topic. What I'm getting at is you should only really be looking at going authentic with something if you believe that your audience has knowledge of the topic. Unless you personally feel the need to.
In the case of occultism, I'd say research 'sacrifices' and make that authentic. Anything else you can either draw some inspiration from real world occultism or make it up completely. Note that when I say sacrifices I'm not including resurrection, I believe that resurrection is something that varies from cult to cult, so there's no real need for it to be authentic. However, you may disagree and want to have a lot of the cult workings authentic, so I'm providing some material to look at:
I found various other ones in my research, but all of them were either fictional creations or people disgusted at cults (not really explaining them, rather degrading them). Interestingly, or rather logically, people seem to avoid the subject all-together. it's very hard to find a reputable source of actual cult in-goings, and no I don't believe its because cults are very secretive (although it could very well be).
What I could suggest is going onto the cult wikipedia page and looking for tangents. It'd be your best bet of finding cults, but it'll also be very time consuming.
Let me wrap up by repeating what I said previously, reason is one of the things that makes a game less scary, so don't worry about how much sense it makes. Just focus on what you want it to be like.
Side note: I think it'd be cool if the game starts where you wake up and you're chained to a stone table with a ritual being performed around you. You see dead body's strewn around the place as you look around, before you look down on the table. The tables completely covered in blood. Then you frantically look up to see a knife plunge into you and you black out again. Next time you wake up you're among the dead bodies, and the ritual is still being performed with more people. You were fortunate enough for them to have missed your heart, but to be physically dead for a few seconds. This is your chance to escape.
I had a chat with my mother who did some research on occultism. She said to read between the lines. Think of what's posted online as a net. A lot of it is their in order to find new recruits. They're seeking people crazy enough to join their cult, so they post things similar to what their cult actually practices in order to find people interested in it.
But besides that, their is not much you can find online. These cults truly believe in what they're doing and are very close-mouthed about it, you're either with them or dead. Your best bet would be to join a cult itself, but I highly recommend you do not. Not only would it be extremely dangerous, if you couldn't stomach it you'd be found out immediately.
Side Note: She said that as she has not had first-hand experience in a cult she cannot guarantee this, its possible that some cults let their members leave on the condition that they do not reveal any information. But in that case, you wouldn't find anything out form them anyways.
She suggested that your best bet would be to go to a library and look up certain key phrases like 'summoning' or 'sacrifice. A lot of the information of cults are old historical documents detailing certain practices that happened in their time. If you fail to find much in your local library, it might be worth travelling to a larger library.
Also, this is a forum focused specifically on occultism that I found: http://www.occultforum.org/forum/index.php?sid=eb9514ea4248849b599cf6993b146041
It'd be a gold mine for real cults to grab fanatics, so I'd say there is a lot of material their that originates from real cults.