Your assessment is correct. You have distinguished "bust-outs" from "eliminations" in a way the TD has historically NOT done, but that is a reasonable way to describe it: Bust-outs are anytime a player loses all of his/her chips and is busted-out of the tournament, and an elimination is a the bust-out where the player does not return to the tournament (by rebuying).
The TD awards bounty chips on bust-outs. It does not distinguish bust-outs from eliminations in this regard.
So if Ted buys a bounty chip and is busted out by Bob, Ted gives his bounty chip to Bob. It doesn't matter if Ted then rebuys. Bob still keeps Ted's bounty chip. However, Ted can buy another bounty chip with his rebuy (if the tournament is configured that way).
There is an open feature request to incorporate both methods. There's a small issue (sort of) where the tournament status would be "wrong" for a time after a player is busted out who then later rebuys. What I mean is that the hitman would receive the bounty chip of the busted player, but when the busted player rebuys the bounty chip would be returned. In our example above, Bob would get Ted's bounty chip, and the tournament status would reflect this (such as Bob's winnings), but when Ted rebuys the chip would be returned to Ted and the tournament status would be updated again. It's a bit janky and would feel like a bug, but there's not really any way around it.
In the meantime, there is a way to workaround this. It's a tiny chore (well, in a tournament of 9 players), but it works. After the tournament, edit each player, go down to the History section, and if a player is busted-out and rebuys at any time during the tournament, simply edit their buy-in and remove the bounty chip amount (in your case, set the $5.00 to 0), and then edit the final rebuy and set the bounty chip amount to $5. Do this for all players.
I just did this for your tournament and it took me about 60 seconds. I see the winnings after doing so are exactly what you expected.
** Edit: One last thing: there's a Bounty Chip column on the Players tab that shows the bounty chip amount paid upon buy-in. This will no longer reflect the correct amount, since anyone who rebuys will have their bounty chip amount removed from the buy-in and added to the rebuy. No matter though, this column is from long ago before bounty chips were even added to rebuys. Best to ignore it altogether and use the other bounty columns.