(This is not to take away from what was previously posted I just want to share my research as I like to get familiar with these types of systems Tibia adds. )
I'd say the reward system and the reward chest go hand in hand. Any monster who has the special loot bag has the reward system. But like mentioned, it all depends on the boss. I was doing some research for hours so I'd thought I'd include what I found out from reading tons of threads and news. Here's a list with the bosses that participate in the reward system from my understanding:
https://tibia.fandom.com/wiki/Reward_Chest
On August 14 2013 Tibia.com posted a teaser, "Reward to Whom Reward is Due" implementing a new reward system for certain bosses:
https://www.tibia.com/news/?subtopic=newsarchive&id=2486
This is the thread in which Cipsoft answered some FAQ about the Reward System:
https://www.tibia.com/forum/?action=thread&postid=35297344%20#post35297344
Q: How does the participation system work?
A: Please note that we will not reveal any details of the participation system. The more you participate in the fight (dealing damage, receiving damage) the higher your individual participation will be.
Q: Do we have to be in a "party" to activate the new reward system?
A: No, the system automatically notices if you participated in the fight or not.
Q: What happens with rare items like Ferumbras' hat?
A: Items with a 100% drop chance, such as Ferumbras' hat, will only drop once and will be awarded to one player with a sufficient participation. Other rare items, such as a great shield, will stay rare and there will only be a chance that it will be dropped but no guarantee. Please note, that we will not reveal more details since it depends on the loot and the boss.
Q: Is there a guarantee that everyone who participated in the fight will be rewarded?
A: No, but the more you took part in the fight, the better is your chance of a reward. That means as well, the more you took part in the fight, the better your reward will be.
https://www.tibia.com/forum/?action=thread&postid=35297733#post35297733
Rares will stay rare. In this case, there is a chance that one player will get a great shield but there is no guarantee that it will drop anyway.
https://www.tibia.com/forum/?action=thread&postid=35304973#post35304973
For example: 10 players reach a certain participation level, then one of the ten players will get the hat but that does not have to be the highest level or the highest damage dealer. It is randomly one out of the ten players who reached the certain participation level.
Here is the actual update thread w/ comments.
https://www.tibia.com/forum/?action=thread&threadid=4088372
On August 26, 2013:
https://www.tibia.com/news/?subtopic=newsarchive&id=2507&fbegind=13&fbeginm=1&fbeginy=2000&fendd=12&fendm=1&fendy=2020&flist=11111111
Of course, the new reward system will work for all new bosses as well.
On Dec 12 2019- Most recently there's been some changes with participation:
https://www.tibia.com/news/?subtopic=newsarchive&id=5323&fbegind=1&fbeginm=1&fbeginy=2000&fendd=11&fendm=1&fendy=2020&flist=11111111
A small change in regards to participation has also been introduced: From now on you have to do damage in a bossfight in order to receive participation, so e.g. healing your teammates is no longer enough. Please note that you have to actually do damage to the boss, with one point being already enough, however. If the boss's resistances fully negate the damage, your attack will not count towards participation.
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