Both are group conversations in Briar, but they grow differently.
This site lists forums only. The bot is a member of each one, which is why it can invite you.
Briar has no server: your phone and the bot talk directly, over Tor, which is slower than the usual messengers. These delays are normal:
/join: a few minutes, while Briar is open
and connected to the bot.In your contact list, a dot next to each contact turns green while Briar is connected to them. The same dot shows at the top of a conversation. When the bot's dot is green, messages, invitations and forum posts can go through; otherwise they wait until it turns green.
The icons at the bottom of Briar's side menu show how Briar connects to your contacts: through the Internet (always over Tor), Wi-Fi or Bluetooth. Tap them to see each one's state, or to turn one on.
If the dot stays off, check that Briar is allowed to run in the background: how to stay connected.
Briar adds a contact at a distance only once both people have entered the other's link: that is how each side knows who it is talking to, without any server in between. The bot never sees your link otherwise, so you send it here, and you enter the bot's link in Briar.
Sharing your link is safe: someone who has it can only ask to become your contact, and nothing happens unless you enter their link too.