That process can of course be optimized in a number of ways that drastically cut down on the network overhead as compared to the naive MxN write pattern, but what's written here is not safe on its own.
In my layman understanding: Given a set, a quorum is some method to choosing a sub set, such that any two such sub sets will always have at least one overlapping member.
Majority is one quorum algorithm - given a set [A,B,C], the majorities are: [A,B,C], [A,B], [A,C] and [B,C]. Any two of those sets will have at least one member overlapping.
However, majority is somewhat wasteful, because the latency of these quorum-based algorithms are almost always bound by the slowest member of the quorum - the more machines you need to wait for, the more likely one of them will be outlier-slow.
You'd potentially be better off choosing a quorum algorithm that requires less than a majority - because that'd mean, in the best case, fewer responses to wait for, lowering the probability that one of those members will be very slow. There are drawbacks to this - it makes fault tolerance and provisioning harder to calculate - but it's got some cool potential benefits.
Some cool ones to explore here: https://pdfs.semanticscholar.org/a243/7f18205414f6398b29c4f8...
The one non-majority quorum commit protocol that most people are probably already familiar with is the "sloppy quorum" replication in Dynamo systems[1] (e.g. Cassandra, Riak, Voldemort, etc.). Basically, since the quorum is configurable on a per-cluster basis instead of being inherent to the protocol, and usually isn't a majority of the cluster, the system can still make progress when half of the nodes are unreachable. (But of course, as the paper notes, this means that you need to resolve conflicts some other way, which adds a whole bunch of complexity.)
1: http://www.allthingsdistributed.com/files/amazon-dynamo-sosp...
Assuming you've chosen correctly between CP and AP approaches, this tells us that availability and latency aren't as important as consistency. But there's nothing that says they aren't arbitrarily close...
Actually, invoking CAP probably didn't add to my message. What I meant to say is that people don't talk about non-majority quorum commits that much because the interesting part is that the serializability comes with majority/overlapping quorums.