As far as I can tell, the game modes for each game go like this:
Halo: CE
Main: Slayer, CTF, KotH, Race, Oddball
Significant Spin-Off/Variants: Assault (CTF Variant), Juggernaut (Oddball Variant)
DLC: None
Total Major Game Modes: 7
Halo 2
Main: Slayer, CTF, Oddball, KotH, Juggernaut, Territories, Assault
Significant Spin-Off/Variants: SWAT
DLC: None
Total Major Game Modes: 8
Halo 3
Main: Slayer, CTF, Oddball, KotH, Juggernaut, Infection, VIP, Territories, Assault
Significant Spin-Off/Variants: SWAT, Race, Grifball
DLC: None
Total Major Game Modes: 12
Halo: Reach
Main: Assault, CTF, Headhunter, Infection, Invasion, Juggernaut, KotH, Oddball, Race, Stockpile, Territories, Slayer
Significant Spin-Off/Variants: Invasion Slayer, Grifball, SWAT
DLC: None
Total Major Game Modes: 15
Halo 4
Main: Slayer, CTF, Oddball, KotH, Dominion, Extraction, Flood, Regicide, Grifball
Significant Spin-Off/Variants: SWAT, Race (Custom, Eventually Offical Gametype)
DLC: Ricochet
Total Major Game Modes: 11
Halo 5: Guardians
Main: Slayer, CTF, Breakout, Strongholds, Warzone, Warzone Assault
Significant Spin-Off/Variants: SWAT
DLC: Assault, Grifball, Fiesta (as standalone game type), Ricochet (through customs), Oddball (through customs)
Total Major Game Modes: 9( Offical Modes only), 11(including possible ball modes like Oddball/Ricochet), 12(with Fiesta)
Obviously, what a person counts as a significant gametype is a matter of opinion, but Halo 5 isn’t necessarily as bad off as people make it out to be. It’s also worth noting that a lot of the Reach-only modes like Headhunter and Stockpile were barely touched and played.