Thanks for jumping in with suggestions. It's actually a harder problem than it seems! I have a feeling "provider-obscured" and "homomorphic" (while accurate!) are significantly less accessible to end users. I'll try to break it down:

Do end users talk about the services they purchase using the word "provider"? Is for example, Facebook or Twitter commonly referred to as a social media "provider?" The most common example I can think of is ISP, but I rarely hear non technical people say "ISP". They say something like "I get Internet from Time Warner" instead.

Is "obscured" a commonly used, highly accessible word? Can you think of a few popular movies or books with that word in the title? Is it commonly used in news headlines?

So the proposal is a hyphenated phrase of these two uncommon words. I think it's likely "Zero Knowledge" would crush "Provider-Obscured" in an A/B test. Ditto for "Homomorphic."

Seeing highly technical terms in headings makes non technical people believe that the software is complex and hard to use, and is therefore not for them. IMO, this is one of the classic failings of security products in general. It needs more study.

Thanks for the explanation of your thinking there. FWIW, I don't think (contra the other posters) that it's that much of a stretch (of standard terminology) to call this "Zero knowledge" -- you are, after all, preventing information from flowing in a certain direction, just like in the ZK proofs.

With that said, what about "opaque" instead of "obscured" and "host" or "cloud" instead of "provider"?

Host-Opaque Cloud Storage

Cloud-Obscured Cloud Storage (okay, bad acronym)

(And I know it's kind of late for a name change anyway.)

You do realize that he is talking about terminology for the general public? The names you are suggesting are just terrible. They don't sound good and are not even close to widely understood terms (no offense intended). Perhaps you are not a native English speaker?

Well said and I encourage you to continue promoting your product with this strategy. Being overly precise with lingo will result in a worse world where only domain experts understand what's going on.