Even though Username changes not propagated to chat on change was fixed, it seems the display name changes are not propagated fully to chat. I changed my display name to "Keelan" a long time ago across the network, but in the chat transcript my old name is still visible. This includes starred posts, as can be seen in this room.
Unlike posts or comments on the main site, messages in chat are always attributed to the name of the author at the time the message was posted. This has a few advantages that are especially useful in the context of chat conversations:
Chat messages referencing or addressing other users by name (that is, not replying to specific messages) still make sense in context days, months, even years later.
Ambiguity is reduced: it's only possible to @-ping a user who is currently or recently active in the room, so chances are if there's an @-ping for a given name then that name will also appear nearby in the transcript (even if many other users have had similar names in the past or future).
There's a certain permanence to what you write in chat that reflects the actual behavior of the system (public, indexed by search engines) and differs (intentionally!) from that of many similar chat systems (IRC, Slack). The goal here is to encourage people to be a bit mindful of what they say, since once it's out there on the 'Net there's nothing we can do to take it back.
This chat system was created with the express goal of supplanting other systems that allowed or subtly encouraged trolling, harassment, and cliquish behavior by otherwise-courteous members of these sites; to this end, it imposes a level of transparency and accountability that is (even now) rare. That said, there are often good reasons to not leave a trail of your past musings on the 'Net - so if for whatever reason you're not comfortable with the public, immutable nature of chat... Then you should avoid using it.
That's not a bug. Chat transcripts, @-addresses in chat, and @-addresses in comments aren't updated on a name change. Posts and comments are because they're tied to a user object in the database, so those names are just links and (I assume) the question page gets the current name from the user table. Updating the (string, not user-object) copies in other places on every name change sounds like a pretty expensive operation. I doubt SE will ever do it globally, but if your situation is particularly problematic I suggest asking the team if they can do anything to help you.