- Chat Owner (can be manager if you want, but I like the idea of having "chat" in front of it)
Keep in mind the #1 rank (what is currently "owner") has full access to chat settings, and the ability to add/remove users from the #2 rank (what is currently "admin"). So your proposal of "Chat Owner" being above "Owner" would be very confusing, if the only difference is a "chat" prefix. If someone makes a ticket and says "How can I let someone who also owns the website have full control over the chat?", explaining that they need to make them a "Chat Owner" is very confusing if the user is already a regular owner.
I'm also not sure if Manager should be above Owner -- I can't think of any scenario, real-world or online, where Manager controls the Owner.
- Chat Owner/Manager (swapping around admin)
Are you saying to leave it as is? The point is that several tickets have come in where users thought "Admin" would be the highest rank & give them total control over the chat. Other newcomers can't tell if the Owner is more powerful than Admin (they don't know that user list is sorted by rank).
This is why I love discussions like these. You'd think it'd be an easy fix to just come up with a different name, but there's a lot of thought process that has to go into it! ;)