I am confused about what the difference between these chat settings is. Fireteam chat makes sense that it would be the people in your fireteam but what is team chat? If I am playing alone, there is nobody else in a team with me at that time, so what is the word team referring to in this context? Does team just mean the other people running around the planet with me? If so, thats a bad way to name it.
I think the naming comes from the PVP game modes where fireteam is the group of people that you decide to group up with in a fireteam and then team is the people actually on the team with you in the game. So if its a 4v4 match, the 3 other people on the team with you but not in your fireteam are the people who you can speak with in team chat.
In PvE, I think this only applies to matchmaking games. You and a buddy are in a fireteam together. You decide to go and do a strike and matchmaking will add a third player to team up with. That third person is not in your fireteam but they are on your team. So i think that is how it applies in this situation.