You probably would want element.
I think conduit's page is doing the right thing. It doesn't really assume you know what matrix is, but rather assumes you're smart enough to click on a link if you don't know.
> Users from every Matrix homeserver can chat with users from all other Matrix servers.
This statement lead me to assumed the "matrix specification" discussed was about server-to-server communication... I'm not interested in digging into the underpinnings of this server-to-server communication (right now) just like I'm not terribly interested in how blockchains keep in sync across nodes... But I am interested in how this whole stack works, and I was lost with this page as the starting point.
So I think it's safe to add an extra sentence to explain the relationship while still keeping the page simplistic and focused.
EDIT: realizing that the landing page may have been updated with new content itself since your original post.
What is Matrix?
Matrix is an open network for secure and decentralized communication. Users from every Matrix homeserver can chat with users from all other Matrix servers. You can even use bridges to communicate with users outside of Matrix, like a community on Discord.
And I see no installation information.
Installation is downloading a binary, setting up autostart and writing/copying a 10 line configuration.toml.
Matrix is a chat protocol. It works like chatting over GIT (distributed databases) and is more like a team messenger like Mattermost or Zulip. Conduit is a server software and not for end users.
More information about Matrix: https://www.freie-messenger.de/en/matrix
if at all, Element is this.
Matrix doesn't really care about what type of users the clients target