assistant2: Adjust elements in the message editor (#27696)

Most notable change in this PR is the changing the default profiles'
names to just "Write" and "Ask". Everything else is mostly
design-related. Here's how it looks like:

<img
src="https://github.com/user-attachments/assets/791948c9-2d63-4523-9d54-08b63a00be6a"
width="600" />

Release Notes:

- N/A

crates/icons/src/icons.rs

@@ -225,6 +225,7 @@ pub enum IconName {
     Unpin,
     Update,
     UserGroup,
+    UserRoundPen,
     Visible,
     Wand,
     Warning,