Destiny 2 – Digital Deluxe Items

There is a list of the badges and what they mean in game which is under Crucible in the Triumphs screen.

Here is a list of them collected by the playing community:

• Best Served Cold - Killed an enemy player who killed you
• Blood for Blood - Killed an enemy player who just killed one of your allies
• Cold Fusion - Go on a Fusion rifle spree.
• Combined Fire - Kill an enemy player by shooting at them as a team.
• Counter Attack - Get a kill while being the last guardian standing.
• Crushed Them - Win a match with a large margin of victory.
• Barricade Breaker - Killed an enemy Titan and destroy their barricade.
• Dodge This - Killed an enemy Hunter during their dodge animation.
• Rift Breaker - Killed an enemy Warlock during their rift phase.
• Dominant Advantage - Score five Advantage or Power Play kills before the opponent recaptures a zone.
• Pitch Perfect - Defeat an opponent with Hammer of Sol at a distance greater than 30 meters.
• Everything Looks Like a Nail - Defeat three opponents within a single Hammer of Sol activation.
• Regent - Killed two enemy Guardians with a sword.
• Annihilation - Land final blows on the entire enemy team before any of them respawn.
• Double Play - Kill 2 enemy players in rapid succession.
• Triple Play - Kill 3 enemy players in rapid succession.
• Lights Out - Kill 4 enemy players in rapid succession.
• We Ran Out of Medals - In a single life, defeat 20 opposing Guardians.
• Fight Me! - Deal the most damage in a match.
• Flag Bearer - Secure the most capture zones during the match
• Reclaimer - Recapture a zone within 15 seconds of it being captured by your opponents.
• Gangs All Here - All allies must survive at the end of a Countdown round.
• Hawkeye - In a single life, defeat two opponents with precision Hand Cannon final blows.
• Hurricane - Kill three enemy Guardians in a single Arcstaff super activation.
• I Live Here Now - Hold two or more zones for at least 1 minute.
• Lethal Instinct - Defeat an opponent within 2 seconds of activating Golden Gun.
• Lightning Strike - Defeat an opponent within 3 seconds of activating Arc Staff.
• Never In Doubt - Win a match in which your team never trailed.
• Handful of Bullets - Kill three enemy Guardians in a single Golden Gun super.
• Not So Fast, My Friend - Defeat an opposing Guardian using your Super while their Super is active.
• Direct Hit - Kill an enemy player instantly with one shot from a Grenade Launcher.
• Power Overwhelming - As a team, defeat all 4 opposing Guardians at least once during a single Power Play.
• Quick Strike - Get the first kill in a game (to be confirmed).
• Ruthless - Perform a 5-kill streak.
• Unyielding - Perform a 10-kill streak.
• Shutdown - Kill an enemy player who is on a kill streak.
• Splash Damage - Kill 2 or more enemy players with a Rocket Launcher.
• The Cycle - In a single match, land at least one final blow with each class of weapon (Kinetic, Energy, Power) and ability (Melee, Grenade, Super).
• Wrecking Crew - As a team, defeat 7 opposing Guardians without any of your team dying.
• Lethal Cadence - In a single match, defeat 7 opponents with Pulse Rifle final blows.
• Undertaker - Land all knockout blows on the opposing team in a single round.
• Time and a Half - Win a match in overtime.
• Undefeated - Complete a match in which you are never defeated by an opponent.
• Sub machinist - In a single life, defeat 2 opponents with Submachine Gun final blows.
• One for Each of You - In a single life, defeat two opponents with precision Hand Cannon final blows.
• Field Scout - In a single match, defeat 5 opponents at long range with Scout Rifle final blows.
• Striker Special - In a single activation, defeat two opponents with Shoulder Charge, then a third with Fists of Havoc.
• Absolute Force - Defeat two or more opponents in a single Fists of Havoc slam.
• Flying Fortress - Defeat an opponent with a Shield Rush within 3 seconds of defeating an opponent with a Sentinel Shield melee.
• Perfect Guard - Block fatal damage within 2 seconds of activating Ward of Dawn.
• Longbow - Defeat an opponent with Shadowshot at a distance greater than 30 meters.
• Falling Star - Defeat an opponent with Brimstone while Daybreak is active.
• Defying Gravity - In a single Daybreak activation, defeat two or more opponents without touching the ground.
• Singularity - Defeat an opponent with a Nova Bomb Vortex.
• From Downtown - Defeat two or more opponents with a Nova Bomb that was in the air for at least 5 seconds.
• Thunderstruck - Defeat an opponent with Landfall while casting Stormtrance.
• Lightning Storm - Defeat two or more opponents in a single Stormtrance activation.
• Entangled - Defeat a tethered opponent within 5 seconds of casting Shadowshot.

Thanks to Timmy Jim for the link to the Destiny 2 Wiki

Reference to medal details from the game's API.

I haven't tried doing it with only 4 but with my experience doing it normal I don't believe it's possible. I've tried doing it multiple times after one person has fallen off and keep going with only 5 and it will end up stopping and you can't continue.

There are multiple videos showing how it's possible to skip parts or to do it solo, by using swords and supers to jump the gaps between reactors, but these will only get you to the last area with nothing happening. You have to get to the last plate on each section and have the "reactor temperature rises" message appear for each to actually progress and have the enemies spawn at the final area.