Rework typing notifications

Previously, the bot only had rudimentary typing notification support.

It used to send a single notification when starting a long task
and did not bother with notifications anymore.
By default matrix-rust-sdk gives these notifications a validity of 4
seconds, so it would expire shortly. If the bot takes longer to respond,
you'd see the typing notification expire and wonder if a response is
coming.

Another edge case is the bot sending an answer quicker and the typing
notice still being on. Some clients (like element-web) seem to hide the
typing notice when a new message comes, so they don't experience this as
problematic.

The reworked typing notification system should be robust:

- typing notices are sent continuously, until the bot finishes doing
  work
- if the bot is performing multiple actions in a room (even for
  different people), typing notices would continue to be sent until the
  bot becomes idle
- as soon as the bot becomes idle, a "not typing anymore" notice is sent
  to clear the state

src/bot/messaging.rs

@@ -317,6 +317,13 @@ impl Messaging {
 
         let start_time = std::time::Instant::now();
 
+        let _typing_notice_guard = self
+            .bot
+            .matrix_link()
+            .rooms()
+            .start_typing_notice(message_context.room())
+            .await;
+
         let event_span = tracing::error_span!("message_controller", ?controller_type);
 
         crate::controller::dispatch_controller(&controller_type, &message_context, &self.bot)