No, you are right, Matrix is a protocol. Some of its major points are:

• It's decentralized (There are homeservers, but federation is built into the protocol. Think of this like email servers. There are many different servers out there, and they can talk to each other. Or not, depending of the configurations the server admins set.)
• It's a interoperable protocol, allowing users to communicate across different service providers and platforms. This is why Beeper used it as it's core.
• • Bridges allows taking posts from one service and posting it to another, and vice versa. For example, users from Discord, IRC, Telegram, etc can all talk between each other with little effort from whatever platform they choose. They don't need to be using a matrix server themselves, just someone who has set up a bridge with one in their spaces.
• • Puppeting allows users using a matrix homeserver & a bridge that supports puppeting to log in with their other service's account. For example, if you were to do this with Discord , you'd then be able to have all over your messages, including dms, bridge to your "matrix" account. You'd also be able respond via your account directly. That is, if you send a message to Discord from your matrix homeserver, it'll get posted via your Discord account and not the Discord bot used for the normal bridging stuff. NOTE: Puppeting is against Dscord's TOS, but is use in this example as most know what it is.
• End-to-end encryption is built heavily into the protocol. Not every space has to have it, but it's a core feature and enabled by default for DM's.
• It's an open source protocol

Personally, I feel there's still a few cons:

• Is is not polished.
• The new user onboarding needs work
• Some QOL things are missing. Thought this is more an issue with Element, the default reference client. They have a very shrill default notification sound and no way to easily change it. There's an open issue with a fair few users who understandable want to be able to change it easily.
• The spec is evolving still, and there's some things that haven't been figured out yet.

In saying that, I do still use Matrix and run a server myself. I think it's good, but needs some work still.

As to a NZ homeserver or space, I'm currently unaware of any sorry. And GoT felt like it was going of the rails before the last season >.<

Matrix does have some good documentation though, you'll find a lot of info on their site here: https://matrix.org