Using a message broker with end user clients directly

I want to find a way to implement push messages from a server to multiple end user clients with the same message.

One of the options I found was to use a message broker and use it to implement the pub/sub pattern. What I'm not sure about is what supposed to be considered a consumer in such a scenario.

What I thought that the general architecture is when using a message broker is:

End user clients <----> Message broker <---> Server (The clients and server can also speak to one another on things that are not related to topic messages)

And the process I at least that is supposed to happen is this:

1) The end user client registers to a specific topic by sending an initial message directly to the message broker.

2) The server got a message about a topic which he wants every end user will get, so it adds a message to the topic.

3) The message broker instantly sends the message to all the end users by itself without the usage of other push message services like SignalR, Pusher etc. (without the usage of them meaning that it might use it behind the scene, but the developer doesn't actually program the sending of the messages).

After that I heard that the consumer is not supposed to be end user clients, but other servers?

Is my description of the process correct? Or is it something else?